WEBVTT 00:00:00.000 --> 00:00:09.920 PROFESSOR: Everyone seems to be here. 00:00:09.920 --> 00:00:12.610 At least all the seats are filled up, so 00:00:12.610 --> 00:00:13.860 why don't we begin. 00:00:17.330 --> 00:00:21.090 One thing that I did finally, to be specific, I told you 00:00:21.090 --> 00:00:24.250 that we would have three quizzes and that they would be 00:00:24.250 --> 00:00:30.240 uniformly spaced one third, and 2/3 and 2.89 thirds of the 00:00:30.240 --> 00:00:31.910 way through the term. 00:00:31.910 --> 00:00:35.520 And I figured out where those dates would be. 00:00:35.520 --> 00:00:38.140 So since I have them, let me give you the date. 00:00:38.140 --> 00:00:50.090 So Quiz 1 is going to be on Thursday, October 6. 00:00:50.090 --> 00:01:04.860 Quiz 2 will be on Tuesday, November 8, and Quiz 3 will be 00:01:04.860 --> 00:01:09.372 on Thursday, December 8. 00:01:12.150 --> 00:01:16.380 And this, as advertised, is exactly one third of the way 00:01:16.380 --> 00:01:17.650 through the lectures. 00:01:17.650 --> 00:01:21.050 This is exactly 2/3 of the way through the lectures, and this 00:01:21.050 --> 00:01:24.660 is the next to the last meeting that we'll have. 00:01:24.660 --> 00:01:28.370 This is the week before the final week of the terms. so I 00:01:28.370 --> 00:01:32.330 deliberately kept it a little bit away from the usual 00:01:32.330 --> 00:01:34.430 end-of-term crunch. 00:01:34.430 --> 00:01:38.780 Being person who is sensitive to symmetry and order, I find 00:01:38.780 --> 00:01:41.220 it highly regrettable that we cannot have the 00:01:41.220 --> 00:01:43.000 quiz on October 8. 00:01:43.000 --> 00:01:46.920 Then you would on the 8th of October, November, December, 00:01:46.920 --> 00:01:47.650 you have a quiz. 00:01:47.650 --> 00:01:53.190 Unfortunately, October 8, much as that would beautify the 00:01:53.190 --> 00:01:58.780 schedule, is on a Saturday, so we can't do it so. 00:01:58.780 --> 00:02:00.660 I could do it, but I don't think anybody would come. 00:02:03.640 --> 00:02:07.480 I've been, believe it or not, going through these problem 00:02:07.480 --> 00:02:08.960 sets in great detail. 00:02:08.960 --> 00:02:13.860 I didn't want to hand them out until I had the list-- 00:02:13.860 --> 00:02:16.260 and you may or may not know-- that is made up for the 00:02:16.260 --> 00:02:19.410 registrants in each class, And we get a nice array of 00:02:19.410 --> 00:02:22.210 photographs with names and department numbers and years 00:02:22.210 --> 00:02:24.160 numbers under them. 00:02:24.160 --> 00:02:27.250 That something, by the way, that only the instructor in a 00:02:27.250 --> 00:02:30.170 class requests, not even the secretary can do it. 00:02:30.170 --> 00:02:32.500 So this is not widely broadcast. 00:02:32.500 --> 00:02:38.090 I found to my dismay though when I finally got it that 00:02:38.090 --> 00:02:40.650 about one third of the pictures are missing. 00:02:40.650 --> 00:02:44.480 And I don't know whether you folks have been photographed 00:02:44.480 --> 00:02:47.580 fairly late on the term, but a third of the 00:02:47.580 --> 00:02:48.520 pictures are missing. 00:02:48.520 --> 00:02:53.410 I already been putting some faces in my memory banks, but 00:02:53.410 --> 00:02:57.290 to match up names and faces so that I can hand out the 00:02:57.290 --> 00:03:00.170 problem sets individually, I need more photographs. 00:03:00.170 --> 00:03:02.470 So I'll do it one way or another next time, and we'll 00:03:02.470 --> 00:03:04.710 see some photographs, additional photographs, have 00:03:04.710 --> 00:03:06.160 appeared in the interim. 00:03:14.940 --> 00:03:18.060 I've given you three problem sets so far, and they were 00:03:18.060 --> 00:03:22.120 really just for fun as well as to get you thinking about the 00:03:22.120 --> 00:03:25.193 right sorts of things, but there were little puzzle. 00:03:25.193 --> 00:03:28.870 I'm, unfortunately, going to have to tell you from now on 00:03:28.870 --> 00:03:30.930 no more cutesy little puzzles. 00:03:30.930 --> 00:03:33.050 You're going to get real problem sets, 00:03:33.050 --> 00:03:36.100 long, tedious, drudgery. 00:03:36.100 --> 00:03:38.250 They're optional though, so that's the saving grace. 00:03:38.250 --> 00:03:42.410 So I'm going to pass it around the problem set 4. 00:03:42.410 --> 00:03:45.340 And this asks you to demonstrate some of the things 00:03:45.340 --> 00:03:48.250 that we talked about last time and convince yourself that 00:03:48.250 --> 00:03:52.020 they really work and ask you to make some simple 00:03:52.020 --> 00:03:55.550 applications of Miller-Bravais indices to 00:03:55.550 --> 00:03:57.910 two-dimensional lattices. 00:03:57.910 --> 00:04:03.220 And then finally, the last problem is to get you thinking 00:04:03.220 --> 00:04:04.330 about patterns. 00:04:04.330 --> 00:04:07.630 And I've asked you to identify the lattice points and the 00:04:07.630 --> 00:04:09.245 symmetry elements in two patterns. 00:04:12.100 --> 00:04:16.515 I'm almost willing to wager that not one of you will get 00:04:16.515 --> 00:04:20.490 it completely right, that there's going to be some 00:04:20.490 --> 00:04:24.190 little thing that you missed or did wrong. 00:04:24.190 --> 00:04:28.480 At one time, I bet a bag of potato chips for everybody in 00:04:28.480 --> 00:04:32.290 the class against the class wagering against me one bag of 00:04:32.290 --> 00:04:32.890 potato chips. 00:04:32.890 --> 00:04:35.060 Well, I'm going to watch my weight, and I've won that 00:04:35.060 --> 00:04:39.730 invariably, and I really can't eat that greasy sort of stuff. 00:04:39.730 --> 00:04:42.620 In any case, you'll see that in the case of symmetry and 00:04:42.620 --> 00:04:47.050 patterns it is immensely simplified when you know 00:04:47.050 --> 00:04:51.100 exactly what to look for, when you know the number of 00:04:51.100 --> 00:04:53.120 possibilities was finite,. 00:04:53.120 --> 00:04:57.110 And when we're through with this, you can ask just one 00:04:57.110 --> 00:05:00.450 question about a pattern and then know exactly what the 00:05:00.450 --> 00:05:03.120 symmetry of that pattern has to be and exactly where to 00:05:03.120 --> 00:05:05.750 look for everything else. 00:05:05.750 --> 00:05:09.250 I can guarantee you that by the time you finish this class 00:05:09.250 --> 00:05:12.700 you will never sit in a bathroom and stare at the tile 00:05:12.700 --> 00:05:15.700 floor and see it in exactly the same way again. 00:05:20.030 --> 00:05:24.190 I think that's a good observation because I like to 00:05:24.190 --> 00:05:27.270 think that that's what education is all about, not 00:05:27.270 --> 00:05:30.390 sitting in the bathroom, but seeing things differently 00:05:30.390 --> 00:05:31.920 after you've had the experience 00:05:31.920 --> 00:05:32.850 than you did before. 00:05:32.850 --> 00:05:36.690 That's what education is all about. 00:05:36.690 --> 00:05:38.580 So have fun with the patterns. 00:05:38.580 --> 00:05:42.260 And again, it will become-- 00:05:42.260 --> 00:05:43.510 one seat up here. 00:05:52.750 --> 00:05:57.300 Today then we're about ready to embark on an adventure. 00:05:57.300 --> 00:06:00.330 As I said last time, we'll develop two-dimensional 00:06:00.330 --> 00:06:04.470 symmetries first because there are relatively few of them, 00:06:04.470 --> 00:06:06.205 and one could do this rigorously 00:06:06.205 --> 00:06:08.510 and in great detail. 00:06:08.510 --> 00:06:11.420 We're going to have to touch on things more lightly when we 00:06:11.420 --> 00:06:13.860 get to three-dimensional symmetries so we can finish at 00:06:13.860 --> 00:06:18.270 a decent point in the term. 00:06:18.270 --> 00:06:27.680 Let me remind you that we have so far identified two types of 00:06:27.680 --> 00:06:32.030 symmetry that can exist in a lattice, onefold, twofold, 00:06:32.030 --> 00:06:36.550 threefold, or sixfold rotation axes. 00:06:36.550 --> 00:06:38.730 Any number of different rotational symmetries are 00:06:38.730 --> 00:06:41.640 possible, but if you're going to want them to be compatible 00:06:41.640 --> 00:06:48.200 with a lattice, you must restrict the rotation axis to 00:06:48.200 --> 00:06:51.710 one of these five, including a onefold axis, which is no 00:06:51.710 --> 00:06:53.320 symmetry it all. 00:06:53.320 --> 00:06:56.150 And in two-dimensions, besides translation, we saw the 00:06:56.150 --> 00:06:59.210 operation of reflection. 00:06:59.210 --> 00:07:02.370 And now we're going to begin to put things together and 00:07:02.370 --> 00:07:05.470 make elaborate combinations. 00:07:05.470 --> 00:07:09.530 The first thing I will ask is can we have more than one of 00:07:09.530 --> 00:07:13.470 these symmetry elements present and operating about 00:07:13.470 --> 00:07:17.084 the same locus at the same time. 00:07:17.084 --> 00:07:19.600 The answer to that would be, why not. 00:07:19.600 --> 00:07:25.310 Because if we look at a sixfold rotation axis, that's 00:07:25.310 --> 00:07:31.100 really what a twofold rotation axis does combined with what a 00:07:31.100 --> 00:07:33.080 threefold rotation axis does. 00:07:39.560 --> 00:07:42.680 And we plot these on top of one another, and what we end 00:07:42.680 --> 00:07:48.120 up with is the arrangement of motifs that is generated by 00:07:48.120 --> 00:07:49.330 sixfold axis. 00:07:49.330 --> 00:07:55.160 So in a sense, a sixfold axis is a twofold axis superimposed 00:07:55.160 --> 00:07:57.410 on a threefold axis. 00:07:57.410 --> 00:08:00.460 That kind of a naive way about thinking of this though. 00:08:00.460 --> 00:08:05.510 What we're really saying is that what a sixfold axis has 00:08:05.510 --> 00:08:18.690 is the set of operations 1 A 2 pi/ 6, A twice 2 pi /6 -- 00:08:18.690 --> 00:08:21.190 that would be 120 degree rotation-- 00:08:21.190 --> 00:08:29.840 and A pi, A 5 pi/6. 00:08:29.840 --> 00:08:32.350 And there's 1, 2, 3, 4, 5. 00:08:32.350 --> 00:08:33.624 And we're missing one. 00:08:33.624 --> 00:08:38.470 60, 120, 240. 00:08:38.470 --> 00:08:50.680 So that would be A twice 2 pi/3, 4 pi/6 00:08:50.680 --> 00:08:54.350 So what we're really saying is a sixfold axis consists of 00:08:54.350 --> 00:08:59.760 these six elements, and these elements constitute a group 00:08:59.760 --> 00:09:06.010 because I can combine any two of these rotation operations 00:09:06.010 --> 00:09:09.440 and find something that's already a member of the set. 00:09:09.440 --> 00:09:13.350 By saying that a sixfold axis consists of a threefold axis 00:09:13.350 --> 00:09:16.400 sitting on top of a twofold axis, what we're saying is 00:09:16.400 --> 00:09:19.630 that there's one collection of elements here, namely 00:09:19.630 --> 00:09:27.430 identity, or same thing as A 2 pi and A pi, that is a subset 00:09:27.430 --> 00:09:29.705 of these, or we could say a subgroup. 00:09:32.870 --> 00:09:35.930 So what we're doing in saying that these two sets of 00:09:35.930 --> 00:09:41.040 rotation operations exist simultaneously, we're saying 00:09:41.040 --> 00:09:45.470 that the twofold axis is a subgroup of the sixfold axis. 00:09:45.470 --> 00:09:49.160 And similarly, we can separate out three other elements, a 00:09:49.160 --> 00:09:59.470 onefold axis, same as A 2 pi; a 120-degree rotation, A 2 00:09:59.470 --> 00:10:08.360 pi/3; and a 240-degree rotation, A 4 pi/3. 00:10:08.360 --> 00:10:13.770 And these three operations, a group of rank 2, is what 00:10:13.770 --> 00:10:20.530 constitutes another subgroup, a subset of elements which by 00:10:20.530 --> 00:10:26.980 themselves satisfy all of the requirements of a group. 00:10:26.980 --> 00:10:32.360 So this is one example of how we can have more than one 00:10:32.360 --> 00:10:35.730 symmetry element, a set of operations existing about the 00:10:35.730 --> 00:10:37.550 same locus and space. 00:10:37.550 --> 00:10:44.640 This is not how one would go about the deriving these 00:10:44.640 --> 00:10:46.110 higher symmetry however. 00:10:46.110 --> 00:10:50.090 Because what we have to do is to add something to the set, 00:10:50.090 --> 00:10:53.680 something that's called an extender, and then show that 00:10:53.680 --> 00:10:58.150 all the requirements of a set being a group is satisfied; 00:10:58.150 --> 00:11:03.190 namely, that a combination of any two elements in the group 00:11:03.190 --> 00:11:07.030 is also a member the group, that for every operation an 00:11:07.030 --> 00:11:10.840 inverse exists, and the identity operation is a member 00:11:10.840 --> 00:11:11.570 of the group. 00:11:11.570 --> 00:11:14.670 So you can build up more higher symmetries, more 00:11:14.670 --> 00:11:18.270 complex symmetries by adding some operation called an 00:11:18.270 --> 00:11:21.680 extender and then taking all of the products of these 00:11:21.680 --> 00:11:27.120 elements and see what new operations arise. 00:11:27.120 --> 00:11:30.840 So we can have more than one symmetry element operating 00:11:30.840 --> 00:11:35.080 about the same locus and the set of individual operations 00:11:35.080 --> 00:11:37.480 that it embodies. 00:11:37.480 --> 00:11:42.970 And here sitting all by itself is a mirror plane. 00:11:42.970 --> 00:11:50.210 Why don't we combine a mirror plane with the rotation axis? 00:11:50.210 --> 00:11:52.210 And we've already seen examples of that. 00:11:52.210 --> 00:11:57.350 For example, in the square pattern that these tiles make 00:11:57.350 --> 00:12:00.220 up, there's a fourfold axis in the middle of each of the 00:12:00.220 --> 00:12:02.440 squares that are also mirror planes that pass 00:12:02.440 --> 00:12:04.200 through that location. 00:12:04.200 --> 00:12:06.570 So let me look at a combination that is a little 00:12:06.570 --> 00:12:08.410 simpler to handle. 00:12:08.410 --> 00:12:13.860 And what I'm going to say is here sits a mirror plane with 00:12:13.860 --> 00:12:17.300 a reflection operation that I'll call sigma 1 for the 00:12:17.300 --> 00:12:19.640 individual operation. 00:12:19.640 --> 00:12:23.790 And let me say that I combine now in that space a second 00:12:23.790 --> 00:12:27.390 reflection operation about a line that 00:12:27.390 --> 00:12:30.780 intersects the first one. 00:12:30.780 --> 00:12:34.610 And the question I'm going to ask now is happens when I take 00:12:34.610 --> 00:12:41.240 a first motif, number 1, reflected in the locus of the 00:12:41.240 --> 00:12:45.180 first mirror plane to get number 2, and then reflect 00:12:45.180 --> 00:12:49.455 that a second time in the locus of the second mirror 00:12:49.455 --> 00:12:52.100 plane to get one that sits up here. 00:12:54.800 --> 00:13:00.360 So the question is now what is the operation sigma 1 followed 00:13:00.360 --> 00:13:03.530 by the operation sigma 2 equals to? 00:13:08.260 --> 00:13:12.750 We can almost answer by the process of elimination. 00:13:12.750 --> 00:13:16.650 If this one is left handed, reflection changes the 00:13:16.650 --> 00:13:20.080 polarity, so number 2 is right hand. 00:13:20.080 --> 00:13:22.970 And if we reflect a second time, the right-handed one 00:13:22.970 --> 00:13:25.510 goes to a left-handed one. 00:13:25.510 --> 00:13:28.770 So what we're asking is how do I get from one left-handed 00:13:28.770 --> 00:13:33.210 motif to another left-handed motif? 00:13:33.210 --> 00:13:36.850 Of the three operations that exist in a two-dimensional 00:13:36.850 --> 00:13:39.640 space could be translation. 00:13:39.640 --> 00:13:42.860 But, clearly, I can't take the first one and slide it 00:13:42.860 --> 00:13:46.240 parallel to itself and make it coincide with the third. 00:13:46.240 --> 00:13:49.490 So translation would not change chirality, but that 00:13:49.490 --> 00:13:51.090 won't work. 00:13:51.090 --> 00:13:53.070 What's left? 00:13:53.070 --> 00:13:53.370 Rotation. 00:13:53.370 --> 00:13:54.310 Yes. 00:13:54.310 --> 00:13:56.110 You want to clarify something first? 00:13:56.110 --> 00:13:58.745 AUDIENCE: That third one, it looks kind of left-handed. 00:13:58.745 --> 00:13:59.020 [INAUDIBLE]. 00:13:59.020 --> 00:13:59.390 PROFESSOR: Oops. 00:13:59.390 --> 00:13:59.810 It does. 00:13:59.810 --> 00:14:00.050 It does indeed. 00:14:00.050 --> 00:14:01.050 Sorry about that. 00:14:01.050 --> 00:14:03.350 When you've got your nose poked right in these things, 00:14:03.350 --> 00:14:04.340 it's easy to overlook it. 00:14:04.340 --> 00:14:05.650 Yes. 00:14:05.650 --> 00:14:06.570 Absolutely right. 00:14:06.570 --> 00:14:08.390 That should be over canted over a bit more like I had it 00:14:08.390 --> 00:14:09.010 the first way. 00:14:09.010 --> 00:14:11.190 Very good thank you. 00:14:11.190 --> 00:14:12.270 I don't want to proceed much. 00:14:12.270 --> 00:14:12.580 Yes? 00:14:12.580 --> 00:14:15.710 AUDIENCE: What did you [INAUDIBLE] 00:14:15.710 --> 00:14:17.120 axis [INAUDIBLE]? 00:14:17.120 --> 00:14:19.220 PROFESSOR: We could do that. 00:14:19.220 --> 00:14:22.970 And let's do it, and we're going to encounter 00:14:22.970 --> 00:14:24.050 this sooner or later. 00:14:24.050 --> 00:14:30.240 So we have a first mirror plane reflect from a first 00:14:30.240 --> 00:14:34.050 one, which is right handed, to second one that's left handed. 00:14:34.050 --> 00:14:39.710 And then define a second locus, and that one would take 00:14:39.710 --> 00:14:43.020 the second one and move it over to here, the same 00:14:43.020 --> 00:14:45.970 distance on the other side of this mirror plane. 00:14:45.970 --> 00:14:51.430 And I'll let you answer the question since you asked it. 00:14:51.430 --> 00:14:53.720 How is the first one related to the third one? 00:14:53.720 --> 00:14:54.480 AUDIENCE: Translation. 00:14:54.480 --> 00:14:54.890 PROFESSOR: Translation. 00:14:54.890 --> 00:14:55.790 Yes. 00:14:55.790 --> 00:14:56.525 AUDIENCE: It's translated. 00:14:56.525 --> 00:15:03.010 PROFESSOR: Exactly So if this distance is delta, the first 00:15:03.010 --> 00:15:05.155 and the third are related be a translation 00:15:05.155 --> 00:15:07.800 which is twice delta. 00:15:07.800 --> 00:15:11.020 So that would be in addition where we'll eventually want to 00:15:11.020 --> 00:15:14.200 consider when we have patterns that are based on a lattice. 00:15:14.200 --> 00:15:19.160 But right now, I want to consider symmetry operations 00:15:19.160 --> 00:15:22.940 that leave at least a point in space and move perhaps a line 00:15:22.940 --> 00:15:25.730 in the case of the mirror plane. 00:15:25.730 --> 00:15:29.130 But back to the original question, what is this first 00:15:29.130 --> 00:15:33.020 reflection followed by a second reflection? 00:15:33.020 --> 00:15:34.610 We've just thrown out translation. 00:15:34.610 --> 00:15:36.970 That won't work unless the mirror planes are parallel to 00:15:36.970 --> 00:15:37.460 one of those. 00:15:37.460 --> 00:15:40.220 The only thing left is rotation. 00:15:40.220 --> 00:15:44.380 And in point of fact, even looking at this diagram, you 00:15:44.380 --> 00:15:50.010 can get from the first one to the third one by a rotation. 00:15:50.010 --> 00:15:53.930 So the answer to the question is that a combination of two 00:15:53.930 --> 00:15:59.200 reflection operations is a rotation operation about the 00:15:59.200 --> 00:16:02.740 point of intersection of the mirror planes. 00:16:02.740 --> 00:16:05.940 But we can go further and say exactly how large this 00:16:05.940 --> 00:16:07.440 rotation is. 00:16:07.440 --> 00:16:09.580 Let's draw an extra line in here. 00:16:09.580 --> 00:16:12.610 And let's say that this angle is alpha. 00:16:12.610 --> 00:16:14.790 And if I repeat this by reflection, that 00:16:14.790 --> 00:16:18.410 angle is also alpha. 00:16:18.410 --> 00:16:20.860 Let's label this angle beta. 00:16:20.860 --> 00:16:24.630 Repeat that by reflection, and that angle must also be beta. 00:16:24.630 --> 00:16:31.360 The angle between the two mirror planes is the angle mu. 00:16:31.360 --> 00:16:36.860 And clearly mu is equal to alpha plus beta. 00:16:36.860 --> 00:16:41.130 So the answer to the question quite generally is have two 00:16:41.130 --> 00:16:45.410 mirror planes intersect at an angle mu, and their successive 00:16:45.410 --> 00:16:49.200 operation is going to be equivalent to the net result 00:16:49.200 --> 00:16:52.050 of rotating through twice of the angle 00:16:52.050 --> 00:16:55.580 between them mirror line. 00:16:55.580 --> 00:16:57.600 So we don't have to think about that anymore now. 00:16:57.600 --> 00:16:59.250 We know if we combine to mirror planes 00:16:59.250 --> 00:17:00.500 we get a net rotation. 00:17:03.980 --> 00:17:09.420 So that is a basic result, and that's a first example of 00:17:09.420 --> 00:17:12.160 something that I'm going to call by my own pet term. 00:17:12.160 --> 00:17:14.859 I'm going to call them combination theorem, the 00:17:14.859 --> 00:17:18.190 theorem that tells you what you get when you combine two 00:17:18.190 --> 00:17:19.380 operations. 00:17:19.380 --> 00:17:23.079 And what this is going to give us is a way of filling in one 00:17:23.079 --> 00:17:25.160 box in the group multiplication table. 00:17:28.730 --> 00:17:35.860 So knowing what two mirror planes combined at an angle 00:17:35.860 --> 00:17:42.500 should be, I can now combine the mirror planes at angles 00:17:42.500 --> 00:17:45.660 which gives me a rotation operation which is one of the 00:17:45.660 --> 00:17:49.010 ones that's allowed for a lattice. 00:17:49.010 --> 00:17:51.500 In general, if I'm not yet putting these symmetry 00:17:51.500 --> 00:17:54.850 elements in a lattice, I could combine them in such a way 00:17:54.850 --> 00:17:58.940 such that I got a 17-fold rotation as the angle. 00:17:58.940 --> 00:18:01.540 And then the angle between the mirror planes would 00:18:01.540 --> 00:18:04.810 be half of 2 pi/17. 00:18:04.810 --> 00:18:08.170 Could be a lovely symmetry, nice for a pendant or a ring, 00:18:08.170 --> 00:18:10.820 but not for a lattice and not for a crystal. 00:18:10.820 --> 00:18:13.130 So I'm going to want to combine mirror planes at 00:18:13.130 --> 00:18:19.040 angles that correspond to rotations that are compatible 00:18:19.040 --> 00:18:21.100 with the lattice. 00:18:21.100 --> 00:18:24.660 So what I could do is take a first reflection 00:18:24.660 --> 00:18:26.695 locus, sigma 1. 00:18:26.695 --> 00:18:32.280 And if I want the rotation angle to be 180 degrees for a 00:18:32.280 --> 00:18:38.550 twofold axis, I would want the second mirror plane with the 00:18:38.550 --> 00:18:45.100 operation sigma 2 to be at one half of pi with 00:18:45.100 --> 00:18:47.230 respect to the first. 00:18:47.230 --> 00:18:52.190 And for this specific combination then, reflection 00:18:52.190 --> 00:18:59.130 in the first mirror planes followed by reflection in the 00:18:59.130 --> 00:19:04.480 second mirror planes to give me a third one, is going to be 00:19:04.480 --> 00:19:05.500 equivalent to-- 00:19:05.500 --> 00:19:07.160 Oops. 00:19:07.160 --> 00:19:08.390 Back here-- 00:19:08.390 --> 00:19:11.065 it's going to be equivalent to a twofold axis. 00:19:15.010 --> 00:19:22.460 So sigma 1 followed by sigma 2 at an angle one half of pi is 00:19:22.460 --> 00:19:25.010 going to be equivalent to the rotation operation A pi. 00:19:38.850 --> 00:19:42.990 This is not yet a complete pattern because here's a 00:19:42.990 --> 00:19:46.395 mirror planes that wants to reflect this motifs as well. 00:19:49.300 --> 00:19:55.500 So let's just reflect number 3 across the mirror planes 1. 00:19:55.500 --> 00:19:58.390 And then we're going to get a fourth object. 00:19:58.390 --> 00:20:01.880 Now let's show that this constitutes a group. 00:20:01.880 --> 00:20:04.100 We've got the identity operation doing nothing. 00:20:09.850 --> 00:20:12.490 We've got the operation sigma 1. 00:20:12.490 --> 00:20:15.730 We've got the operation sigma 2, and we've got 00:20:15.730 --> 00:20:18.820 the operation a pi. 00:20:18.820 --> 00:20:23.380 I'll put the same operations again in the vertical column. 00:20:23.380 --> 00:20:27.310 Here's operation 1 doing nothing, sigma 1, 00:20:27.310 --> 00:20:29.360 sigma 2, and A pi. 00:20:32.050 --> 00:20:36.090 So doing nothing and following it by one of these four 00:20:36.090 --> 00:20:40.600 operations just gives you the same thing back again. 00:20:40.600 --> 00:20:44.790 And doing these operations followed by 1 will give me the 00:20:44.790 --> 00:20:46.535 same operation back again. 00:20:49.300 --> 00:20:54.940 If I reflect twice across sigma 1 from left to right and 00:20:54.940 --> 00:20:57.690 right to left, I'm back to where I started from. 00:20:57.690 --> 00:21:03.250 So sigma 1 followed by sigma 1 is the identity operation. 00:21:03.250 --> 00:21:08.150 Sigma 1 followed by sigma 2 is what we just did. 00:21:08.150 --> 00:21:11.520 That turns out to be the operation A pi. 00:21:11.520 --> 00:21:16.800 And sigma 1 reflection and following that by A pi is the 00:21:16.800 --> 00:21:18.890 way in which I get to the fourth one. 00:21:18.890 --> 00:21:25.210 So that is going to be equal to sigma 2. 00:21:25.210 --> 00:21:27.320 Well, let me rips through the last ones fairly quickly. 00:21:27.320 --> 00:21:31.060 Sigma 2 followed by signal 1. 00:21:31.060 --> 00:21:34.720 Sigma 2 followed by sigma 1 is the same as A pi. 00:21:37.620 --> 00:21:40.590 Sigma 2 followed by sigma 2 gets us back to where we 00:21:40.590 --> 00:21:41.220 started from. 00:21:41.220 --> 00:21:43.400 That's the identity operation. 00:21:43.400 --> 00:21:48.290 Sigma 2 followed by A pi reflect from here to here and 00:21:48.290 --> 00:21:49.290 then rotate. 00:21:49.290 --> 00:21:53.150 That's the same as sigma 1. 00:21:53.150 --> 00:21:57.090 And the final sequence, rotation followed by 00:21:57.090 --> 00:21:59.930 reflection would give me sigma 2. 00:21:59.930 --> 00:22:03.430 Rotation followed by sigma 2 will give me sigma 1. 00:22:03.430 --> 00:22:07.120 And doing the rotation operation twice is the same as 00:22:07.120 --> 00:22:09.890 the identity operation. 00:22:09.890 --> 00:22:11.823 So are the group postulates satisfied? 00:22:14.700 --> 00:22:16.270 Yes. 00:22:16.270 --> 00:22:25.190 The combination of any two elements is a 00:22:25.190 --> 00:22:26.440 member of the group. 00:22:37.840 --> 00:22:40.813 For every operation, an inverse exists. 00:22:50.000 --> 00:22:55.670 And we can answer that question very easily by merely 00:22:55.670 --> 00:22:59.950 looking at the column under a particular element, and 00:22:59.950 --> 00:23:03.700 somewhere we find the identity operation. 00:23:03.700 --> 00:23:08.090 So sigma 1 is its own inverse. 00:23:08.090 --> 00:23:11.690 So, yes, an inverse exists for every operation. 00:23:11.690 --> 00:23:14.715 And then finally, identity is a member of the group. 00:23:20.240 --> 00:23:22.430 So everything's lights up. 00:23:22.430 --> 00:23:24.260 Bells ring. 00:23:24.260 --> 00:23:26.940 This set of four elements is entitled to 00:23:26.940 --> 00:23:30.680 call itself a group. 00:23:30.680 --> 00:23:36.490 And in particular, it is a group of rank 4 because there 00:23:36.490 --> 00:23:39.370 are four elements in the group. 00:23:39.370 --> 00:23:40.900 I show you a cool thing. 00:23:40.900 --> 00:23:45.560 I noticed that the number of objects in the pattern, the 00:23:45.560 --> 00:23:48.550 number of motifs in the pattern, is exactly the same 00:23:48.550 --> 00:23:50.480 as the order of the group. 00:23:50.480 --> 00:23:51.520 Why? 00:23:51.520 --> 00:23:56.390 Because these four operations tell you how to get from any 00:23:56.390 --> 00:24:01.500 operation to the remaining three and tell you how to get 00:24:01.500 --> 00:24:02.260 it into itself. 00:24:02.260 --> 00:24:04.290 So identity relates it to itself. 00:24:04.290 --> 00:24:06.080 Sigma 1 reflects it to here. 00:24:06.080 --> 00:24:08.190 A pi rotates it down to here. 00:24:08.190 --> 00:24:10.740 Sigma 2 reflects it down to here. 00:24:10.740 --> 00:24:14.300 So there's always a one-to-one correspondence between the 00:24:14.300 --> 00:24:17.770 elements that are in the group and the rank of a group and 00:24:17.770 --> 00:24:21.050 the number of objects that are in the pattern. 00:24:21.050 --> 00:24:21.380 Yes, sir? 00:24:21.380 --> 00:24:23.970 AUDIENCE: How do you show if the inverse exists again? 00:24:23.970 --> 00:24:29.660 PROFESSOR: The inverse exists if I can find something that 00:24:29.660 --> 00:24:34.170 combined with an element in my basic set gives me the 00:24:34.170 --> 00:24:36.192 identity operation. 00:24:36.192 --> 00:24:39.860 So it's the operation followed by its inverse has to be the 00:24:39.860 --> 00:24:41.630 same as doing nothing. 00:24:41.630 --> 00:24:44.530 So if I look under each element and find the identity 00:24:44.530 --> 00:24:48.655 operation, then sigma 2 is its own inverse. 00:24:48.655 --> 00:24:52.680 If I look under A pi, A pi is its own inverse. 00:24:52.680 --> 00:24:53.970 This is generally not the case. 00:24:53.970 --> 00:24:55.535 This is a very simple symmetry. 00:25:04.420 --> 00:25:07.250 So really we can work with commas and little figures, and 00:25:07.250 --> 00:25:08.970 that's one way of getting these symmetries. 00:25:08.970 --> 00:25:12.050 But group theory I think you can see is a very nice, very 00:25:12.050 --> 00:25:15.840 elegant language for describing some 00:25:15.840 --> 00:25:17.135 characteristics of these patterns. 00:25:19.640 --> 00:25:20.890 Now little bit of jargon. 00:25:23.485 --> 00:25:27.370 As if this material were not confusing enough, there are 00:25:27.370 --> 00:25:32.780 two different languages that are used to denote these 00:25:32.780 --> 00:25:35.560 unique combinations. 00:25:35.560 --> 00:25:40.530 This notation where we simply have a number for a rotation 00:25:40.530 --> 00:25:45.310 axis or an m for mirror plane is something that was first 00:25:45.310 --> 00:25:49.290 proposed by two 00:25:49.290 --> 00:25:52.170 mathematicians, Hermann and Mauguin. 00:25:55.830 --> 00:26:03.360 And this was adopted as the preferred notation in the 00:26:03.360 --> 00:26:07.660 international tables, that hefty tome that I brought it 00:26:07.660 --> 00:26:08.715 on the first day of classes. 00:26:08.715 --> 00:26:12.410 So this is also referred to synonymously as the 00:26:12.410 --> 00:26:13.660 international notation. 00:26:21.100 --> 00:26:24.370 Let me put these things out horizontally. 00:26:24.370 --> 00:26:30.930 For rotation axes by themselves, the symbol in 00:26:30.930 --> 00:26:34.280 Hermann notation is simply n. 00:26:34.280 --> 00:26:37.710 And for lattice, n-- 00:26:37.710 --> 00:26:40.030 as we've seen many times-- 00:26:40.030 --> 00:26:45.090 is restricted to what in Hermann-Mauguin notation would 00:26:45.090 --> 00:26:49.475 be either 1, 2, 3, 4, and 6. 00:26:52.820 --> 00:26:57.670 There's another type of notation that is use quite 00:26:57.670 --> 00:27:03.630 frequently by condensed matter physicist, and this is called 00:27:03.630 --> 00:27:12.380 the Schoenflies notation after one of the mathematicians who 00:27:12.380 --> 00:27:20.660 was the first to derive the three-dimensional symmetries. 00:27:20.660 --> 00:27:22.130 This was a curious thing. 00:27:22.130 --> 00:27:25.530 There were three people in different parts of the world 00:27:25.530 --> 00:27:27.830 about the turn of the century-- and this was before 00:27:27.830 --> 00:27:30.170 email or even air mail-- 00:27:30.170 --> 00:27:33.520 who were trying to derive the three-dimensional symmetries. 00:27:33.520 --> 00:27:36.450 Schoenflies was one of them. 00:27:36.450 --> 00:27:40.430 And all three of them got to the same place at about the 00:27:40.430 --> 00:27:45.220 same time, and the results were obtained. 00:27:45.220 --> 00:27:47.090 But each one used his own method and had 00:27:47.090 --> 00:27:48.350 his different notation. 00:27:48.350 --> 00:27:51.470 Schoenflies was one of the people who did this. 00:27:51.470 --> 00:27:58.190 Schoenflies was a mathematician and based his 00:27:58.190 --> 00:27:59.990 notation on group theory. 00:28:02.530 --> 00:28:05.190 Something called a cyclic group is what 00:28:05.190 --> 00:28:06.610 the C stands for. 00:28:12.010 --> 00:28:16.170 And a cyclic group is one in which all elements are power, 00:28:16.170 --> 00:28:29.660 so to speak, of some basic operation. 00:28:39.720 --> 00:28:43.390 For example, a fourfold axis consists of the set of 00:28:43.390 --> 00:28:50.010 operations A pi/2; a 90 degree rotation; A pi, which can be 00:28:50.010 --> 00:28:55.160 written as A pi over 2 squared; A pi over 2 squared; 00:28:55.160 --> 00:28:55.750 [? then we need ?] 00:28:55.750 --> 00:29:04.470 A pi/2; and A pi/ 2 again; A 3 pi/2, which can be viewed as 00:29:04.470 --> 00:29:08.660 doing A pi/2 three times; and finally, the identity 00:29:08.660 --> 00:29:13.450 operation, which is the same as A 2 pi, which is equivalent 00:29:13.450 --> 00:29:18.160 to doing the basic operation A pi/2 four times. 00:29:18.160 --> 00:29:21.830 So the group of rank 4, there are four elements in the 00:29:21.830 --> 00:29:26.790 group, each one is a power of the basic operation A pi/2. 00:29:26.790 --> 00:29:30.840 So a rotation axis by itself as a cyclic group. 00:29:30.840 --> 00:29:34.740 Schoenflies indicates these generically by C subscript n, 00:29:34.740 --> 00:29:36.890 where n is the rank of the axis. 00:29:36.890 --> 00:29:38.870 Hence, this is C1. 00:29:38.870 --> 00:29:40.540 This is C2. 00:29:40.540 --> 00:29:42.230 This is C3. 00:29:42.230 --> 00:29:43.400 This is C4. 00:29:43.400 --> 00:29:44.650 And this is C6. 00:29:54.220 --> 00:29:57.670 A mirror plane by itself is indicated m in the 00:29:57.670 --> 00:30:02.300 international notation. 00:30:02.300 --> 00:30:04.810 A mirror plane is also a cyclic group. 00:30:04.810 --> 00:30:08.240 There are two operations, reflection and reflection back 00:30:08.240 --> 00:30:09.290 to where you came from. 00:30:09.290 --> 00:30:11.730 Doing the reflection operation twice is the 00:30:11.730 --> 00:30:13.160 same as doing nothing. 00:30:13.160 --> 00:30:18.990 So Schoenflies also call this one C. And the one thing that 00:30:18.990 --> 00:30:27.520 is without meaning in English is his subscript S. And 00:30:27.520 --> 00:30:33.880 Schoenflies was German, and the S stands for spiegel, 00:30:33.880 --> 00:30:37.020 which is the German word for mirror. 00:30:37.020 --> 00:30:39.170 We have in many cities a paper that's 00:30:39.170 --> 00:30:40.770 called the Daily Mirror. 00:30:40.770 --> 00:30:44.252 In Germany, there's a paper called Der Spiegel. 00:30:44.252 --> 00:30:46.950 So they use it the same way in everyday life 00:30:46.950 --> 00:30:48.620 as well as in notation. 00:30:48.620 --> 00:30:51.840 So C sub S in the Schoenflies notation is a mirror plane. 00:30:51.840 --> 00:30:55.340 Cyclic group the S stand for spiegel. 00:30:55.340 --> 00:31:07.990 For the combinations of which we have seen only one, in the 00:31:07.990 --> 00:31:12.560 Schoenflies notation these are all of the form Cnv, a 00:31:12.560 --> 00:31:16.180 rotation axis Cn with a vertical mirror plane passing 00:31:16.180 --> 00:31:16.940 through it. 00:31:16.940 --> 00:31:19.310 So this one, for example-- we've yet 00:31:19.310 --> 00:31:21.000 to look at the others-- 00:31:21.000 --> 00:31:23.620 would be C2v. 00:31:23.620 --> 00:31:26.170 So we'll have to draw the first, and then I'll give you 00:31:26.170 --> 00:31:28.630 the complete set. 00:31:28.630 --> 00:31:32.410 So this is a rotation axis with a mirror plane passing 00:31:32.410 --> 00:31:35.170 vertically through it. 00:31:35.170 --> 00:31:36.450 Let's do a few more. 00:31:36.450 --> 00:31:42.210 And I think having done a couple in great detail, we'll 00:31:42.210 --> 00:31:46.040 see what the others will look like quite readily. 00:31:46.040 --> 00:31:51.630 Let's take a fourfold axis and add to the rotation operations 00:31:51.630 --> 00:32:00.870 A pi/2, a first mirror plane that I'll label sigma 1. 00:32:00.870 --> 00:32:04.630 I can permute the order of operations and say that sigma 00:32:04.630 --> 00:32:09.430 1 followed by the rotation operation A pi/2 is going to 00:32:09.430 --> 00:32:14.120 be equal to a second reflection operation that is 00:32:14.120 --> 00:32:20.830 equal to one half of pi/2 away from the first. 00:32:20.830 --> 00:32:26.590 So I'm claiming that if I reflect in the first mirror 00:32:26.590 --> 00:32:31.610 plane and then rotate by 90 degrees that should be another 00:32:31.610 --> 00:32:35.010 mirror plane at 45 degrees to the first. 00:32:35.010 --> 00:32:38.110 So let me do exactly what I advertised. 00:32:38.110 --> 00:32:40.150 Here's the first, one right handed. 00:32:40.150 --> 00:32:43.640 Reflect across to get a second one, which is left handed. 00:32:43.640 --> 00:32:50.820 Reflect, then rotate, and that would bring my left-handed one 00:32:50.820 --> 00:32:52.610 up to this location here. 00:32:52.610 --> 00:32:54.860 So here's 3, and it's left handed. 00:32:54.860 --> 00:32:57.700 And lo and behold, just as advertised, I get from the 00:32:57.700 --> 00:33:00.720 first one to the third one by reflecting across a mirror 00:33:00.720 --> 00:33:05.020 plane, which is one half of pi/2. 00:33:09.720 --> 00:33:13.890 If let these symmetry elements operate on each other, what 00:33:13.890 --> 00:33:15.990 I'll end up with is a set of mirror planes 00:33:15.990 --> 00:33:19.090 at 45 degree intervals. 00:33:19.090 --> 00:33:25.960 And what I'll have is a pair of objects hung in the same 00:33:25.960 --> 00:33:28.625 fashion at every mirror plane. 00:33:33.810 --> 00:33:35.550 Is that too fast? 00:33:35.550 --> 00:33:36.870 You want to go through that a little slower? 00:33:40.390 --> 00:33:42.160 Oh, go ahead say, go through it more slowly. 00:33:42.160 --> 00:33:44.810 Everybody's afraid to say, yeah, yeah, and been seem like 00:33:44.810 --> 00:33:45.620 a class dummy. 00:33:45.620 --> 00:33:46.870 Do you want me to do it more slowly? 00:33:51.090 --> 00:33:54.430 I see people still writing, so I think what you'd rather have 00:33:54.430 --> 00:33:57.200 is me be quiet for a bit while you catch up to where I am. 00:34:06.900 --> 00:34:14.679 In the Schoenflies notation, this lovely thing here is C4. 00:34:14.679 --> 00:34:19.860 It's a rotation axis of rank 4 with a vertical mirror plane 00:34:19.860 --> 00:34:21.860 added to it. 00:34:21.860 --> 00:34:25.290 And the international notation, now the 00:34:25.290 --> 00:34:28.929 Hermann-Mauguin notation, is just a running list of the 00:34:28.929 --> 00:34:32.830 individual symmetry elements that are present. 00:34:32.830 --> 00:34:37.770 And now I really are going to have a mouthful. 00:34:37.770 --> 00:34:41.040 This is a fourfold axis, so the symbol for that is 4. 00:34:41.040 --> 00:34:42.150 This is a mirror plane. 00:34:42.150 --> 00:34:43.239 This is a mirror plane. 00:34:43.239 --> 00:34:44.260 This is a mirror plane. 00:34:44.260 --> 00:34:45.190 This is a mirror plane. 00:34:45.190 --> 00:34:46.515 This is a mirror plane. 00:34:46.515 --> 00:34:48.199 This is a mirror plane. 00:34:48.199 --> 00:34:49.820 That's a mirror plane, and that's a mirror plane. 00:34:49.820 --> 00:34:58.750 So it looks as though I should call this 4mmmmmmmm, which 00:34:58.750 --> 00:35:00.000 comes 4mmmmmmmm. 00:35:02.130 --> 00:35:04.460 Well, that is a typical reaction. 00:35:04.460 --> 00:35:06.880 It's a lovely symmetry. 00:35:06.880 --> 00:35:10.120 But that is a mouthful. 00:35:10.120 --> 00:35:14.630 So it isn't really necessary to give all these m's So the 00:35:14.630 --> 00:35:24.370 international notation is a running list of the 00:35:24.370 --> 00:35:28.210 independent symmetry elements, and that's the new wrinkle 00:35:28.210 --> 00:35:31.110 that I'm introducing. 00:35:31.110 --> 00:35:34.305 It's a running list of the independent symmetry elements. 00:35:41.254 --> 00:35:46.570 And all these mirror planes are just different sigma that 00:35:46.570 --> 00:35:49.090 exist as operations in the group. 00:35:49.090 --> 00:35:52.390 But how many different kinds of mirror planes are they? 00:35:52.390 --> 00:35:57.060 Well, there are two different kinds of mirror planes, both 00:35:57.060 --> 00:36:01.630 in terms of the way in which they function in the pattern. 00:36:01.630 --> 00:36:05.380 The two motifs related by reflection hang close to this 00:36:05.380 --> 00:36:09.550 mirror plane, but they're widely separated for this 00:36:09.550 --> 00:36:10.850 mirror plane. 00:36:10.850 --> 00:36:15.100 So the motifs do different things relative to those two 00:36:15.100 --> 00:36:16.630 kinds of mirror planes. 00:36:16.630 --> 00:36:20.420 Another way of asking what's independent is if I start with 00:36:20.420 --> 00:36:25.210 this 1 mirror plane and repeat it by 90-degree rotations, 00:36:25.210 --> 00:36:29.660 I'll get these 4 mirror planes 90 degrees away. 00:36:29.660 --> 00:36:34.950 So they are not independent in that these mirror planes are 00:36:34.950 --> 00:36:39.140 all related by the rotational symmetry that's present. 00:36:39.140 --> 00:36:42.630 You don't get this mirror plane in any fashion other 00:36:42.630 --> 00:36:46.730 than saying, if I combine the rotation operation with this 00:36:46.730 --> 00:36:50.970 reflection operation sigma, the net result is this 00:36:50.970 --> 00:36:52.230 reflection plane. 00:36:52.230 --> 00:36:55.310 So there are two mirror planes that are distinct in this 00:36:55.310 --> 00:36:57.440 arrangement of symmetry elements, distinct in the 00:36:57.440 --> 00:37:01.550 sense that they function in different ways in the pattern; 00:37:01.550 --> 00:37:05.380 distinct in the sense that no other operation that is 00:37:05.380 --> 00:37:10.160 present will throw these two operations into one another. 00:37:10.160 --> 00:37:15.660 Another example, and this is in fact symmetry 4mm, is the 00:37:15.660 --> 00:37:17.300 square tile. 00:37:17.300 --> 00:37:19.670 If you look at the mirror planes there, they are 45 00:37:19.670 --> 00:37:20.730 degrees apart. 00:37:20.730 --> 00:37:23.850 But one of those mirror planes comes out normal to the edge 00:37:23.850 --> 00:37:25.060 of the tile. 00:37:25.060 --> 00:37:28.710 The other mirror plane comes out of the vertex of the tile. 00:37:28.710 --> 00:37:32.390 So they are different in the way they function in the space 00:37:32.390 --> 00:37:34.450 which has this symmetry. 00:37:34.450 --> 00:37:38.110 So, mercifully, we don't have to call this 4mmmmmmmm. 00:37:38.110 --> 00:37:42.750 We drop the last 6 m's, and this one is called simply for 00:37:42.750 --> 00:37:47.360 4mm, 2 kinds of mirror planes with a fourfold axis. 00:37:47.360 --> 00:37:49.990 If you're familiar with the way in which these were 00:37:49.990 --> 00:37:55.780 derived, the symbol tells you exactly what you've got. 00:37:55.780 --> 00:38:00.510 So it's a very useful notation for these symmetries. 00:38:03.770 --> 00:38:06.820 Let me pause here, give you a chance to catch up. 00:38:06.820 --> 00:38:07.302 Yes, sir? 00:38:07.302 --> 00:38:09.471 AUDIENCE: So wait, all you did here is take two separate 00:38:09.471 --> 00:38:12.122 mirror planes at an angle of 45 degrees between, and you 00:38:12.122 --> 00:38:13.770 ended up with a fourfold symmetry. 00:38:13.770 --> 00:38:14.030 PROFESSOR: Exactly. 00:38:14.030 --> 00:38:17.564 AUDIENCE: But you didn't put on the fourfold symmetry, it 00:38:17.564 --> 00:38:19.060 was just what fell out of it? 00:38:19.060 --> 00:38:20.545 PROFESSOR: No. 00:38:20.545 --> 00:38:23.730 I Got this mirror plane to begin with by combining this 00:38:23.730 --> 00:38:27.770 reflection operation sigma with the operation A pi/2, and 00:38:27.770 --> 00:38:29.340 that's where this operation came from. 00:38:34.440 --> 00:38:37.480 But a general theorem that says if I have a rotation 00:38:37.480 --> 00:38:43.020 operation A alpha and combine it with a reflection operation 00:38:43.020 --> 00:38:48.640 sigma, the combined effect is a reflection operation sigma 00:38:48.640 --> 00:38:54.510 2, which is alpha/2 away from the first. 00:38:54.510 --> 00:38:57.360 So, again, showing you for this now rather messy diagram, 00:38:57.360 --> 00:39:01.290 if I start with the reflection operation that takes 1 of them 00:39:01.290 --> 00:39:05.890 throws of the 2 and then rotates 2 up to location 3, 00:39:05.890 --> 00:39:12.510 the way I get from 1 to 3 in one shot is to reflect in this 00:39:12.510 --> 00:39:13.760 locus sigma 2. 00:39:16.420 --> 00:39:16.850 Yes, sir? 00:39:16.850 --> 00:39:18.188 AUDIENCE: You could have started with any one of those 00:39:18.188 --> 00:39:20.232 two and got the third. 00:39:20.232 --> 00:39:21.690 PROFESSOR: Absolutely. 00:39:21.690 --> 00:39:23.250 Absolutely. 00:39:23.250 --> 00:39:27.370 The operations that are present operate 00:39:27.370 --> 00:39:30.090 everything in the space. 00:39:30.090 --> 00:39:33.460 So when I say that there's a mirror plane here that relates 00:39:33.460 --> 00:39:36.340 this one to this one, it also relates this one to this one, 00:39:36.340 --> 00:39:38.845 this one to this one, this one to this one, that mirror 00:39:38.845 --> 00:39:40.720 planes operates on everything. 00:39:40.720 --> 00:39:44.070 You can't say that a symmetry operation grab this little 00:39:44.070 --> 00:39:47.060 packet of space and moves it to another packet 00:39:47.060 --> 00:39:48.360 removed from it. 00:39:48.360 --> 00:39:52.000 To say it's a symmetry of a pattern or of a crystal, it 00:39:52.000 --> 00:39:54.850 has to leave everything invariant. 00:39:54.850 --> 00:39:56.100 OK. 00:39:59.200 --> 00:40:02.630 But, again, in terms of the language of groups theory, if 00:40:02.630 --> 00:40:06.730 you combine this pair of operations, then when you 00:40:06.730 --> 00:40:10.560 combine everything that you get pairwise, you will get in 00:40:10.560 --> 00:40:15.720 this case a total of, how many operations? 00:40:15.720 --> 00:40:19.390 How many operations are in this pattern? 00:40:19.390 --> 00:40:22.570 I said a moment ago the rank of the group is the number of 00:40:22.570 --> 00:40:24.130 operations that are present. 00:40:24.130 --> 00:40:26.860 It's the number of objects that are in the group because 00:40:26.860 --> 00:40:29.040 each operation in a group tells you how to get from 00:40:29.040 --> 00:40:30.520 anyone to all of the other. 00:40:30.520 --> 00:40:34.336 So, 2, 4, 6, 8, this is a group of rank 8, and there are 00:40:34.336 --> 00:40:36.911 8 operation that are present. 00:40:36.911 --> 00:40:39.320 And I can rattle them off quickly. 00:40:39.320 --> 00:40:43.390 Four operations, identity, 90, 180, 270 for 00:40:43.390 --> 00:40:44.340 the fourfold axis. 00:40:44.340 --> 00:40:46.770 That's 4. 00:40:46.770 --> 00:40:48.230 This mirror plane. 00:40:48.230 --> 00:40:50.230 This mirror plane. 00:40:50.230 --> 00:40:51.550 The mirror plane. 00:40:51.550 --> 00:40:52.470 And this mirror plane. 00:40:52.470 --> 00:40:54.130 That's 8. 00:40:54.130 --> 00:40:55.950 Four for rotation. 00:40:55.950 --> 00:40:57.200 Four for reflection. 00:41:06.370 --> 00:41:07.620 Other questions? 00:41:13.205 --> 00:41:13.860 All right. 00:41:13.860 --> 00:41:18.130 Let me then wrap up this quickly and get 00:41:18.130 --> 00:41:19.040 onto something new. 00:41:19.040 --> 00:41:21.715 Did I hear the hiccup or a question? 00:41:21.715 --> 00:41:22.100 No. 00:41:22.100 --> 00:41:25.200 It was a hiccup. 00:41:25.200 --> 00:41:26.450 Not a yawn I hope. 00:41:38.630 --> 00:41:42.830 Let me do the highest symmetry of all in two dimensions, and 00:41:42.830 --> 00:41:47.220 this is if I take a sixfold axis and combine it with a 00:41:47.220 --> 00:41:48.630 mirror plane. 00:41:48.630 --> 00:41:52.030 So here's the first operation sigma. 00:41:52.030 --> 00:41:55.190 This one is something of a bear to draw. 00:41:55.190 --> 00:41:56.700 This is sigma 1. 00:41:56.700 --> 00:42:02.050 Sigma 1 followed by the operation A 2 pi/6 should be 00:42:02.050 --> 00:42:09.840 equal to a reflection that is an angle one half of 2 pi/6 30 00:42:09.840 --> 00:42:11.170 degrees away from the first. 00:42:11.170 --> 00:42:14.250 So this will be sigma 2. 00:42:14.250 --> 00:42:21.530 So if I reflect from 1 to 2 and then rotate by 60 degrees, 00:42:21.530 --> 00:42:24.755 I'm going to get one that sets up here. 00:42:24.755 --> 00:42:29.760 And the way I get from number 1 to number 3 in one shot is 00:42:29.760 --> 00:42:31.570 by a reflection sigma 2. 00:42:31.570 --> 00:42:34.790 Now if I draw in all of those are mirror planes when they 00:42:34.790 --> 00:42:41.220 are repeated by the sixfold axis, I shouldn't have given 00:42:41.220 --> 00:42:43.900 this my 6mmmmmm treatment because there 00:42:43.900 --> 00:42:45.710 are 12 mirror planes. 00:42:45.710 --> 00:42:49.990 Two objects hanging on this one in one fashion spaced in a 00:42:49.990 --> 00:42:52.270 different way that are belly to belly. 00:42:52.270 --> 00:42:55.030 They're back to back here, so it does something different in 00:42:55.030 --> 00:42:56.040 the pattern. 00:42:56.040 --> 00:42:57.430 A pair hanging here. 00:42:57.430 --> 00:43:01.920 A pair hanging in here. 00:43:01.920 --> 00:43:05.262 A pair hanging and here. 00:43:05.262 --> 00:43:06.920 A pair hanging in here. 00:43:09.910 --> 00:43:13.770 And finally, a pair hanging in here. 00:43:13.770 --> 00:43:17.730 So there are a total 2, 4, 6, 8, 10, 12 00:43:17.730 --> 00:43:19.350 motifs in this pattern. 00:43:19.350 --> 00:43:24.900 Pairs hanging on mirror planes that are 30 degrees apart and 00:43:24.900 --> 00:43:28.840 hanging disposed and pointing in different ways on the 00:43:28.840 --> 00:43:30.360 adjacent mirror planes. 00:43:30.360 --> 00:43:33.910 So this is one that we would call in international notation 00:43:33.910 --> 00:43:39.215 6mmm and in Schoenflies notation C6v. 00:43:44.070 --> 00:43:48.270 So we're making extraordinary progress here. 00:43:48.270 --> 00:43:53.370 The one that I did for you initially was C2v, and that's 00:43:53.370 --> 00:43:57.300 2mn in international notation. 00:43:57.300 --> 00:44:02.030 The only one that I left out to this point is a threefold 00:44:02.030 --> 00:44:04.070 axis compared with a vertical mirror planes. 00:44:08.440 --> 00:44:20.370 And let's start with the operation A 2 pi/3, combined 00:44:20.370 --> 00:44:24.050 with that a first reflection operation sigma 1 that passes 00:44:24.050 --> 00:44:25.300 through it. 00:44:27.620 --> 00:44:32.380 And for reference, I'll drawn in some lines that are 00:44:32.380 --> 00:44:36.690 separated by intervals of 60 degrees. 00:44:36.690 --> 00:44:42.610 So this reflection from 1 to 2 followed by a rotation 120 00:44:42.610 --> 00:44:46.740 degrees should give me this one as number 3. 00:44:46.740 --> 00:44:48.280 The first is right handed. 00:44:48.280 --> 00:44:54.360 The second is left handed, and the third stays left handed. 00:44:54.360 --> 00:44:59.950 And lo and behold, the way I get from 1 to 3 directly is by 00:44:59.950 --> 00:45:07.450 a reflection sigma 2 across a mirror line that is-- 00:45:07.450 --> 00:45:08.310 I'm sorry. 00:45:08.310 --> 00:45:13.190 This is number 1 down here-- across a mirror line that is 00:45:13.190 --> 00:45:16.504 30 degrees away from the first. 00:45:16.504 --> 00:45:18.970 If I would complete the pattern, reflect this one 00:45:18.970 --> 00:45:24.840 across to here, take this pair and reflect it or rotate it up 00:45:24.840 --> 00:45:29.440 here, and I have six objects. 00:45:29.440 --> 00:45:31.610 So this is a group of rank 6. 00:45:31.610 --> 00:45:34.980 And it has characteristics that are quite analogous to 00:45:34.980 --> 00:45:39.220 those we did for the other rotation axes in all respect 00:45:39.220 --> 00:45:41.520 except one. 00:45:41.520 --> 00:45:45.915 The international symbol for this combination is C3v. 00:45:48.560 --> 00:45:52.090 We've got a threefold axis. 00:45:52.090 --> 00:45:55.060 We've got the mirror plane that we added. 00:45:55.060 --> 00:45:59.510 And then we've got a mirror plane that is 30 degrees away 00:45:59.510 --> 00:46:00.155 from the first-- 00:46:00.155 --> 00:46:01.310 I'm sorry-- 00:46:01.310 --> 00:46:03.810 60 degrees away from the first, 00:46:03.810 --> 00:46:05.370 one half of 120 degrees. 00:46:10.570 --> 00:46:15.700 I claim that this is not the proper symbol because the 00:46:15.700 --> 00:46:17.710 mirror planes that are listed should be the 00:46:17.710 --> 00:46:22.110 symmetry-independent, distinct sort of mirror planes. 00:46:22.110 --> 00:46:25.890 And is that the case with this one? 00:46:29.110 --> 00:46:29.960 The answer is no. 00:46:29.960 --> 00:46:31.230 It isn't. 00:46:31.230 --> 00:46:32.550 Here is one mirror plane. 00:46:32.550 --> 00:46:34.750 It's got a pair of motifs hanging on it. 00:46:34.750 --> 00:46:38.360 This mirror plane here is a mirror plane that has a pair 00:46:38.360 --> 00:46:40.270 of motifs hanging it. 00:46:40.270 --> 00:46:42.530 Same is true of this mirror plane. 00:46:42.530 --> 00:46:46.880 So all six of these mirror planes are doing the same 00:46:46.880 --> 00:46:48.130 thing in the pattern. 00:46:48.130 --> 00:46:51.020 Each one has a pair of a motifs on either side of it in 00:46:51.020 --> 00:46:52.890 the same fashion. 00:46:52.890 --> 00:46:56.850 And I can get one mirror plane and the motifs hanging on it 00:46:56.850 --> 00:47:00.740 by a rotation of 120 degrees. 00:47:00.740 --> 00:47:03.780 So there's only one kind of mirror planes, so we don't 00:47:03.780 --> 00:47:05.570 need that m. 00:47:05.570 --> 00:47:10.076 So all of the other rotational symmetries, m, a onefold axis 00:47:10.076 --> 00:47:14.090 with a mirror plane; 2mm, 4mm, 6mm have two 00:47:14.090 --> 00:47:16.970 kinds of mirror planes. 00:47:16.970 --> 00:47:21.670 C3v has only one kind of mirror planes. 00:47:21.670 --> 00:47:23.830 And another way of showing that is if I look at a 00:47:23.830 --> 00:47:26.960 trigonal prism. 00:47:26.960 --> 00:47:29.800 It's got a mirror plane coming out of a corner and out of the 00:47:29.800 --> 00:47:31.410 opposite face. 00:47:31.410 --> 00:47:34.580 Mirror plane coming out of the corner and out 00:47:34.580 --> 00:47:35.920 the opposite face. 00:47:35.920 --> 00:47:38.400 Mirror plane coming out of a corner and out 00:47:38.400 --> 00:47:39.750 the opposite face. 00:47:39.750 --> 00:47:41.920 And that's exactly the rearrangement of symmetry 00:47:41.920 --> 00:47:43.030 elements here. 00:47:43.030 --> 00:47:46.620 So each mirror planes when drawn in relative to a 00:47:46.620 --> 00:47:50.560 trigonal prism, which is a body that has this symmetry, 00:47:50.560 --> 00:47:53.510 each mirror planes does exactly the same thing, as 00:47:53.510 --> 00:47:57.340 opposed to the square tile or square prism that has one kind 00:47:57.340 --> 00:48:00.100 of mirror plane that comes out of faces and one kind of 00:48:00.100 --> 00:48:01.850 mirror plane that comes out of corners. 00:48:01.850 --> 00:48:03.500 So there are two different types of mirror planes. 00:48:10.500 --> 00:48:18.740 Let's add them all up, summarize, and we've got the 00:48:18.740 --> 00:48:23.730 cast of characters that we can be use in deriving 00:48:23.730 --> 00:48:25.920 two-dimensional symmetries. 00:48:25.920 --> 00:48:30.650 We've got C1, which is a onefold axis; C2, twofold 00:48:30.650 --> 00:48:36.950 axis; C3, that's a threefold axis; C4, and that's a 00:48:36.950 --> 00:48:42.550 fourfold axis; C6, that's a sixfold axis. 00:48:42.550 --> 00:48:46.690 Then we've got the additions that involve adding a mirror 00:48:46.690 --> 00:48:48.640 plane to a rotational axis. 00:48:48.640 --> 00:48:55.890 So this is simply m and CS in Schoenflies notation. 00:48:55.890 --> 00:48:58.460 Then we added a mirror plane to a twofold axis. 00:48:58.460 --> 00:49:07.160 We've got C2 with a vertical mirror plane, 2mm; C3v; and 00:49:07.160 --> 00:49:18.180 not 3mm, but only 3m; C4v, and that's 4mm; and C6v, and 00:49:18.180 --> 00:49:20.990 that's 6mm. 00:49:20.990 --> 00:49:24.890 Add them all up, and there are 10 unique possibilities, no 00:49:24.890 --> 00:49:26.480 more, no less. 00:49:26.480 --> 00:49:26.650 Yes, sir? 00:49:26.650 --> 00:49:29.090 AUDIENCE: I'm missing a point. 00:49:29.090 --> 00:49:33.810 Why is there the mirror plane sigma 2 rather than rotating 00:49:33.810 --> 00:49:36.820 sigma 1 is not different? 00:49:36.820 --> 00:49:38.290 How do you? 00:49:38.290 --> 00:49:40.690 PROFESSOR: The same thing is going on, on this mirror plane 00:49:40.690 --> 00:49:43.730 as on this mirror planes, on this mirror plane. 00:49:43.730 --> 00:49:48.120 Or if you like to see it in terms of a geometric solid, 00:49:48.120 --> 00:49:52.420 the mirror planes here in a trigonal prism all of them do 00:49:52.420 --> 00:49:53.010 the same thing. 00:49:53.010 --> 00:49:57.360 They come out of one corner and out of the opposite edge. 00:49:57.360 --> 00:50:00.770 Each of these has one pair of objects hanging on it. 00:50:00.770 --> 00:50:04.800 And that is true of all three of them. 00:50:04.800 --> 00:50:11.000 And if you like, this end of the mirror plane here is 00:50:11.000 --> 00:50:15.380 nothing more than one that's related to the first one by 00:50:15.380 --> 00:50:18.550 one of the rotations of the threefold axis extended back 00:50:18.550 --> 00:50:20.340 in the opposite direction. 00:50:20.340 --> 00:50:22.290 So the mirror plane doesn't just work up here. 00:50:22.290 --> 00:50:23.480 It works down here as well. 00:50:23.480 --> 00:50:25.790 It works all through the space. 00:50:25.790 --> 00:50:28.630 So any way you want to look at it and whatever terms work for 00:50:28.630 --> 00:50:32.900 you, each of these mirror planes is the same, but they 00:50:32.900 --> 00:50:35.860 have a different type of end to them. 00:50:35.860 --> 00:50:37.720 There's something different hanging at one end than 00:50:37.720 --> 00:50:41.230 at the other end. 00:50:41.230 --> 00:50:43.510 Think of it in terms of actual physical object. 00:50:43.510 --> 00:50:45.990 AUDIENCE: [? What about the ?] case of the fourfold 00:50:45.990 --> 00:50:46.982 [? there? ?] 00:50:46.982 --> 00:50:51.190 PROFESSOR: In the fourfold, one comes out of the face. 00:50:51.190 --> 00:50:53.210 One comes out at the edge. 00:50:53.210 --> 00:50:58.360 The motifs are oriented and spaced at a different distance 00:50:58.360 --> 00:51:00.005 from each of those neighboring mirror planes. 00:51:09.650 --> 00:51:10.950 So these two are different. 00:51:10.950 --> 00:51:12.970 The fourfold axis never rotates this 00:51:12.970 --> 00:51:15.410 one into this one. 00:51:15.410 --> 00:51:19.790 These two are not different because the threefold axis 00:51:19.790 --> 00:51:25.990 rotates, if you do it twice, this end into this end of what 00:51:25.990 --> 00:51:29.300 is one in the same mirror plane as what's down on here. 00:51:29.300 --> 00:51:32.190 So what's confusing you I think is that there's a 00:51:32.190 --> 00:51:35.170 polarity to the mirror planes. 00:51:35.170 --> 00:51:37.930 Both ends of the mirror planes do not have the same 00:51:37.930 --> 00:51:41.990 disposition of motifs on them, but there's nothing that says 00:51:41.990 --> 00:51:43.650 that this has to be the case. 00:51:43.650 --> 00:51:46.630 The motifs could just be at one end. 00:51:46.630 --> 00:51:50.360 And that, if you think about it a little bit if it help, 00:51:50.360 --> 00:51:51.660 figure out-- 00:51:51.660 --> 00:51:54.180 not on company time though, but on your own time-- what a 00:51:54.180 --> 00:51:56.390 fivefold axis does. 00:51:56.390 --> 00:51:59.350 And a fivefold axis is a non crystallographic symmetry. 00:51:59.350 --> 00:52:09.450 But it turns out that C5v, which is a 5 with an m, has 00:52:09.450 --> 00:52:13.870 only one kind of mirror plane in it as well and for exactly 00:52:13.870 --> 00:52:14.610 the same reason. 00:52:14.610 --> 00:52:17.670 A regular figure that has the symmetry is a pentagon. 00:52:17.670 --> 00:52:20.300 A mirror plane comes out a corner and out of 00:52:20.300 --> 00:52:21.860 the opposite face. 00:52:21.860 --> 00:52:25.210 And that's true for all of the mirror planes that are in 00:52:25.210 --> 00:52:30.000 there and separated by one half of 2 pi/5. 00:52:30.000 --> 00:52:31.920 AUDIENCE: That's just whether or not the [INAUDIBLE] axis is 00:52:31.920 --> 00:52:32.400 [INAUDIBLE]? 00:52:32.400 --> 00:52:32.980 PROFESSOR: Yeah. 00:52:32.980 --> 00:52:33.965 That's what I said. 00:52:33.965 --> 00:52:36.380 Yeah. 00:52:36.380 --> 00:52:40.910 So before I go through all of the rotation axes with 00:52:40.910 --> 00:52:43.577 vertical mirror planes, the international notation would 00:52:43.577 --> 00:52:53.450 be 2mm, 3m, 4mm, 5m, and 6mm, 7m. 00:52:53.450 --> 00:52:55.850 For the odd symmetries, there's only one kind of 00:52:55.850 --> 00:52:59.760 mirror plane in a pattern, in the tile, or 00:52:59.760 --> 00:53:01.890 whatever you want to have. 00:53:01.890 --> 00:53:06.680 So there are 10 distinct possibilities, and these are 00:53:06.680 --> 00:53:09.440 called the Point Groups. 00:53:15.410 --> 00:53:16.200 Why? 00:53:16.200 --> 00:53:21.050 Because they are clusters of symmetry elements about at 00:53:21.050 --> 00:53:23.900 least one point that's fixed and embedded 00:53:23.900 --> 00:53:25.150 immovably in space. 00:53:28.890 --> 00:53:32.150 They're called groups because, as we've seen for one simple 00:53:32.150 --> 00:53:37.620 example, the collection of operations follows the 00:53:37.620 --> 00:53:40.810 postulates for the set of elements which we have defined 00:53:40.810 --> 00:53:42.060 as a group. 00:53:44.710 --> 00:53:47.880 More specifically, we could call these the 10 00:53:47.880 --> 00:53:50.750 two-dimensional crystallographic graphic Point 00:53:50.750 --> 00:54:01.840 Groups, which is more of a mouthful. 00:54:01.840 --> 00:54:04.360 But it emphasizes the fact that there are lots of 00:54:04.360 --> 00:54:07.790 two-dimensional Point Groups, but the two-dimensional 00:54:07.790 --> 00:54:11.470 crystallographic Point Groups are those that involve 00:54:11.470 --> 00:54:14.930 rotational symmetry that are compatible with a lattice. 00:54:14.930 --> 00:54:18.210 Hence, they are crystallographic. 00:54:18.210 --> 00:54:21.950 But the number of Point Groups is actually infinite if you 00:54:21.950 --> 00:54:27.187 include the ones that are not compatible with translation. 00:54:27.187 --> 00:54:28.660 All right. 00:54:28.660 --> 00:54:31.070 I think that's probably a good place to quit. 00:54:31.070 --> 00:54:31.770 And guess what? 00:54:31.770 --> 00:54:34.470 My internal clock has told me that this is the 00:54:34.470 --> 00:54:36.440 time for our break. 00:54:36.440 --> 00:54:42.140 This is one crossroads in our development. 00:54:42.140 --> 00:54:47.180 And what we'll do next for the faint hearted who may not want 00:54:47.180 --> 00:54:49.510 to have more of this stuff for one day than what we've just 00:54:49.510 --> 00:54:54.020 done, we're now going to do the final penultimate 00:54:54.020 --> 00:54:59.790 combination, and say we have also shown that there are 5 00:54:59.790 --> 00:55:01.040 two-dimensional lattices. 00:55:08.710 --> 00:55:11.370 And the final step will be to say if we have a pattern that 00:55:11.370 --> 00:55:15.170 has symmetry and is based on translation, we can obtain 00:55:15.170 --> 00:55:19.530 these by taking each of the 10 crystallographic Point Groups 00:55:19.530 --> 00:55:27.270 in turn and dropping them into each of the five lattices that 00:55:27.270 --> 00:55:28.560 can accommodate them. 00:55:28.560 --> 00:55:32.630 We would not try, for example, to take a Point Group like 4mm 00:55:32.630 --> 00:55:34.880 and try to drop it into the hexagonal lattice. 00:55:34.880 --> 00:55:36.230 It's not going to fit. 00:55:36.230 --> 00:55:41.660 But there will be two, maybe three ways in which we can add 00:55:41.660 --> 00:55:44.250 a given Point Group to one of these lattice types. 00:55:44.250 --> 00:55:47.240 And what we've done when we finished is we have 00:55:47.240 --> 00:55:52.690 exhaustively derived the symmetries' translational 00:55:52.690 --> 00:55:57.010 lattices and symmetry operations of reflection and 00:55:57.010 --> 00:56:01.960 rotation that are possible for two-dimensional crystal. 00:56:01.960 --> 00:56:05.210 You might say, well two-dimensional crystal, I've 00:56:05.210 --> 00:56:08.310 heard that people who do thin film work to make monolayers 00:56:08.310 --> 00:56:09.950 and really make a two-dimensional crystal. 00:56:09.950 --> 00:56:12.710 That's not something you could say only a few years ago. 00:56:12.710 --> 00:56:16.350 But why do we worry about two-dimensional symmetries if 00:56:16.350 --> 00:56:18.840 we're not to be wallpaper designers? 00:56:18.840 --> 00:56:20.910 Well, actually, I'll give you one example. 00:56:20.910 --> 00:56:23.930 One of the difficulties you had in conveying the nature of 00:56:23.930 --> 00:56:26.370 a crystal structure is taking something that's fairly 00:56:26.370 --> 00:56:29.130 complicated in three dimensions and getting it onto 00:56:29.130 --> 00:56:30.790 a two-dimensional sheet of paper. 00:56:30.790 --> 00:56:35.120 And what you invariably do is you project the contents of 00:56:35.120 --> 00:56:38.170 the unit cell down along one of the cell edges. 00:56:38.170 --> 00:56:40.720 And what you have then is a two-dimensional crystal 00:56:40.720 --> 00:56:43.200 structure with a lattice and symmetry. 00:56:43.200 --> 00:56:46.860 So you'll see two-dimensional representations of 00:56:46.860 --> 00:56:49.540 translationally periodic arrangements of atoms quite 00:56:49.540 --> 00:56:52.280 frequently when you look at projections of actual crystal 00:56:52.280 --> 00:56:56.150 structures, which is the only reasonable way to convey the 00:56:56.150 --> 00:56:59.300 information of any structure once you get on beyond the 00:56:59.300 --> 00:57:02.910 baby stuff of sodium chlorides, zinc, sulfide and 00:57:02.910 --> 00:57:04.160 body-centered iron. 00:57:06.670 --> 00:57:08.060 OK. 00:57:08.060 --> 00:57:10.720 I'm going to pause, suck in air. 00:57:10.720 --> 00:57:16.130 I am happy to turn you loose for 10 minutes and will meet 00:57:16.130 --> 00:57:17.790 here again, let's say, at 10 after 3:00.