1 00:00:00,000 --> 00:00:09,920 PROFESSOR: Everyone seems to be here. 2 00:00:09,920 --> 00:00:12,610 At least all the seats are filled up, so 3 00:00:12,610 --> 00:00:13,860 why don't we begin. 4 00:00:17,330 --> 00:00:21,090 One thing that I did finally, to be specific, I told you 5 00:00:21,090 --> 00:00:24,250 that we would have three quizzes and that they would be 6 00:00:24,250 --> 00:00:30,240 uniformly spaced one third, and 2/3 and 2.89 thirds of the 7 00:00:30,240 --> 00:00:31,910 way through the term. 8 00:00:31,910 --> 00:00:35,520 And I figured out where those dates would be. 9 00:00:35,520 --> 00:00:38,140 So since I have them, let me give you the date. 10 00:00:38,140 --> 00:00:50,090 So Quiz 1 is going to be on Thursday, October 6. 11 00:00:50,090 --> 00:01:04,860 Quiz 2 will be on Tuesday, November 8, and Quiz 3 will be 12 00:01:04,860 --> 00:01:09,372 on Thursday, December 8. 13 00:01:12,150 --> 00:01:16,380 And this, as advertised, is exactly one third of the way 14 00:01:16,380 --> 00:01:17,650 through the lectures. 15 00:01:17,650 --> 00:01:21,050 This is exactly 2/3 of the way through the lectures, and this 16 00:01:21,050 --> 00:01:24,660 is the next to the last meeting that we'll have. 17 00:01:24,660 --> 00:01:28,370 This is the week before the final week of the terms. so I 18 00:01:28,370 --> 00:01:32,330 deliberately kept it a little bit away from the usual 19 00:01:32,330 --> 00:01:34,430 end-of-term crunch. 20 00:01:34,430 --> 00:01:38,780 Being person who is sensitive to symmetry and order, I find 21 00:01:38,780 --> 00:01:41,220 it highly regrettable that we cannot have the 22 00:01:41,220 --> 00:01:43,000 quiz on October 8. 23 00:01:43,000 --> 00:01:46,920 Then you would on the 8th of October, November, December, 24 00:01:46,920 --> 00:01:47,650 you have a quiz. 25 00:01:47,650 --> 00:01:53,190 Unfortunately, October 8, much as that would beautify the 26 00:01:53,190 --> 00:01:58,780 schedule, is on a Saturday, so we can't do it so. 27 00:01:58,780 --> 00:02:00,660 I could do it, but I don't think anybody would come. 28 00:02:03,640 --> 00:02:07,480 I've been, believe it or not, going through these problem 29 00:02:07,480 --> 00:02:08,960 sets in great detail. 30 00:02:08,960 --> 00:02:13,860 I didn't want to hand them out until I had the list-- 31 00:02:13,860 --> 00:02:16,260 and you may or may not know-- that is made up for the 32 00:02:16,260 --> 00:02:19,410 registrants in each class, And we get a nice array of 33 00:02:19,410 --> 00:02:22,210 photographs with names and department numbers and years 34 00:02:22,210 --> 00:02:24,160 numbers under them. 35 00:02:24,160 --> 00:02:27,250 That something, by the way, that only the instructor in a 36 00:02:27,250 --> 00:02:30,170 class requests, not even the secretary can do it. 37 00:02:30,170 --> 00:02:32,500 So this is not widely broadcast. 38 00:02:32,500 --> 00:02:38,090 I found to my dismay though when I finally got it that 39 00:02:38,090 --> 00:02:40,650 about one third of the pictures are missing. 40 00:02:40,650 --> 00:02:44,480 And I don't know whether you folks have been photographed 41 00:02:44,480 --> 00:02:47,580 fairly late on the term, but a third of the 42 00:02:47,580 --> 00:02:48,520 pictures are missing. 43 00:02:48,520 --> 00:02:53,410 I already been putting some faces in my memory banks, but 44 00:02:53,410 --> 00:02:57,290 to match up names and faces so that I can hand out the 45 00:02:57,290 --> 00:03:00,170 problem sets individually, I need more photographs. 46 00:03:00,170 --> 00:03:02,470 So I'll do it one way or another next time, and we'll 47 00:03:02,470 --> 00:03:04,710 see some photographs, additional photographs, have 48 00:03:04,710 --> 00:03:06,160 appeared in the interim. 49 00:03:14,940 --> 00:03:18,060 I've given you three problem sets so far, and they were 50 00:03:18,060 --> 00:03:22,120 really just for fun as well as to get you thinking about the 51 00:03:22,120 --> 00:03:25,193 right sorts of things, but there were little puzzle. 52 00:03:25,193 --> 00:03:28,870 I'm, unfortunately, going to have to tell you from now on 53 00:03:28,870 --> 00:03:30,930 no more cutesy little puzzles. 54 00:03:30,930 --> 00:03:33,050 You're going to get real problem sets, 55 00:03:33,050 --> 00:03:36,100 long, tedious, drudgery. 56 00:03:36,100 --> 00:03:38,250 They're optional though, so that's the saving grace. 57 00:03:38,250 --> 00:03:42,410 So I'm going to pass it around the problem set 4. 58 00:03:42,410 --> 00:03:45,340 And this asks you to demonstrate some of the things 59 00:03:45,340 --> 00:03:48,250 that we talked about last time and convince yourself that 60 00:03:48,250 --> 00:03:52,020 they really work and ask you to make some simple 61 00:03:52,020 --> 00:03:55,550 applications of Miller-Bravais indices to 62 00:03:55,550 --> 00:03:57,910 two-dimensional lattices. 63 00:03:57,910 --> 00:04:03,220 And then finally, the last problem is to get you thinking 64 00:04:03,220 --> 00:04:04,330 about patterns. 65 00:04:04,330 --> 00:04:07,630 And I've asked you to identify the lattice points and the 66 00:04:07,630 --> 00:04:09,245 symmetry elements in two patterns. 67 00:04:12,100 --> 00:04:16,515 I'm almost willing to wager that not one of you will get 68 00:04:16,515 --> 00:04:20,490 it completely right, that there's going to be some 69 00:04:20,490 --> 00:04:24,190 little thing that you missed or did wrong. 70 00:04:24,190 --> 00:04:28,480 At one time, I bet a bag of potato chips for everybody in 71 00:04:28,480 --> 00:04:32,290 the class against the class wagering against me one bag of 72 00:04:32,290 --> 00:04:32,890 potato chips. 73 00:04:32,890 --> 00:04:35,060 Well, I'm going to watch my weight, and I've won that 74 00:04:35,060 --> 00:04:39,730 invariably, and I really can't eat that greasy sort of stuff. 75 00:04:39,730 --> 00:04:42,620 In any case, you'll see that in the case of symmetry and 76 00:04:42,620 --> 00:04:47,050 patterns it is immensely simplified when you know 77 00:04:47,050 --> 00:04:51,100 exactly what to look for, when you know the number of 78 00:04:51,100 --> 00:04:53,120 possibilities was finite,. 79 00:04:53,120 --> 00:04:57,110 And when we're through with this, you can ask just one 80 00:04:57,110 --> 00:05:00,450 question about a pattern and then know exactly what the 81 00:05:00,450 --> 00:05:03,120 symmetry of that pattern has to be and exactly where to 82 00:05:03,120 --> 00:05:05,750 look for everything else. 83 00:05:05,750 --> 00:05:09,250 I can guarantee you that by the time you finish this class 84 00:05:09,250 --> 00:05:12,700 you will never sit in a bathroom and stare at the tile 85 00:05:12,700 --> 00:05:15,700 floor and see it in exactly the same way again. 86 00:05:20,030 --> 00:05:24,190 I think that's a good observation because I like to 87 00:05:24,190 --> 00:05:27,270 think that that's what education is all about, not 88 00:05:27,270 --> 00:05:30,390 sitting in the bathroom, but seeing things differently 89 00:05:30,390 --> 00:05:31,920 after you've had the experience 90 00:05:31,920 --> 00:05:32,850 than you did before. 91 00:05:32,850 --> 00:05:36,690 That's what education is all about. 92 00:05:36,690 --> 00:05:38,580 So have fun with the patterns. 93 00:05:38,580 --> 00:05:42,260 And again, it will become-- 94 00:05:42,260 --> 00:05:43,510 one seat up here. 95 00:05:52,750 --> 00:05:57,300 Today then we're about ready to embark on an adventure. 96 00:05:57,300 --> 00:06:00,330 As I said last time, we'll develop two-dimensional 97 00:06:00,330 --> 00:06:04,470 symmetries first because there are relatively few of them, 98 00:06:04,470 --> 00:06:06,205 and one could do this rigorously 99 00:06:06,205 --> 00:06:08,510 and in great detail. 100 00:06:08,510 --> 00:06:11,420 We're going to have to touch on things more lightly when we 101 00:06:11,420 --> 00:06:13,860 get to three-dimensional symmetries so we can finish at 102 00:06:13,860 --> 00:06:18,270 a decent point in the term. 103 00:06:18,270 --> 00:06:27,680 Let me remind you that we have so far identified two types of 104 00:06:27,680 --> 00:06:32,030 symmetry that can exist in a lattice, onefold, twofold, 105 00:06:32,030 --> 00:06:36,550 threefold, or sixfold rotation axes. 106 00:06:36,550 --> 00:06:38,730 Any number of different rotational symmetries are 107 00:06:38,730 --> 00:06:41,640 possible, but if you're going to want them to be compatible 108 00:06:41,640 --> 00:06:48,200 with a lattice, you must restrict the rotation axis to 109 00:06:48,200 --> 00:06:51,710 one of these five, including a onefold axis, which is no 110 00:06:51,710 --> 00:06:53,320 symmetry it all. 111 00:06:53,320 --> 00:06:56,150 And in two-dimensions, besides translation, we saw the 112 00:06:56,150 --> 00:06:59,210 operation of reflection. 113 00:06:59,210 --> 00:07:02,370 And now we're going to begin to put things together and 114 00:07:02,370 --> 00:07:05,470 make elaborate combinations. 115 00:07:05,470 --> 00:07:09,530 The first thing I will ask is can we have more than one of 116 00:07:09,530 --> 00:07:13,470 these symmetry elements present and operating about 117 00:07:13,470 --> 00:07:17,084 the same locus at the same time. 118 00:07:17,084 --> 00:07:19,600 The answer to that would be, why not. 119 00:07:19,600 --> 00:07:25,310 Because if we look at a sixfold rotation axis, that's 120 00:07:25,310 --> 00:07:31,100 really what a twofold rotation axis does combined with what a 121 00:07:31,100 --> 00:07:33,080 threefold rotation axis does. 122 00:07:39,560 --> 00:07:42,680 And we plot these on top of one another, and what we end 123 00:07:42,680 --> 00:07:48,120 up with is the arrangement of motifs that is generated by 124 00:07:48,120 --> 00:07:49,330 sixfold axis. 125 00:07:49,330 --> 00:07:55,160 So in a sense, a sixfold axis is a twofold axis superimposed 126 00:07:55,160 --> 00:07:57,410 on a threefold axis. 127 00:07:57,410 --> 00:08:00,460 That kind of a naive way about thinking of this though. 128 00:08:00,460 --> 00:08:05,510 What we're really saying is that what a sixfold axis has 129 00:08:05,510 --> 00:08:18,690 is the set of operations 1 A 2 pi/ 6, A twice 2 pi /6 -- 130 00:08:18,690 --> 00:08:21,190 that would be 120 degree rotation-- 131 00:08:21,190 --> 00:08:29,840 and A pi, A 5 pi/6. 132 00:08:29,840 --> 00:08:32,350 And there's 1, 2, 3, 4, 5. 133 00:08:32,350 --> 00:08:33,624 And we're missing one. 134 00:08:33,624 --> 00:08:38,470 60, 120, 240. 135 00:08:38,470 --> 00:08:50,680 So that would be A twice 2 pi/3, 4 pi/6 136 00:08:50,680 --> 00:08:54,350 So what we're really saying is a sixfold axis consists of 137 00:08:54,350 --> 00:08:59,760 these six elements, and these elements constitute a group 138 00:08:59,760 --> 00:09:06,010 because I can combine any two of these rotation operations 139 00:09:06,010 --> 00:09:09,440 and find something that's already a member of the set. 140 00:09:09,440 --> 00:09:13,350 By saying that a sixfold axis consists of a threefold axis 141 00:09:13,350 --> 00:09:16,400 sitting on top of a twofold axis, what we're saying is 142 00:09:16,400 --> 00:09:19,630 that there's one collection of elements here, namely 143 00:09:19,630 --> 00:09:27,430 identity, or same thing as A 2 pi and A pi, that is a subset 144 00:09:27,430 --> 00:09:29,705 of these, or we could say a subgroup. 145 00:09:32,870 --> 00:09:35,930 So what we're doing in saying that these two sets of 146 00:09:35,930 --> 00:09:41,040 rotation operations exist simultaneously, we're saying 147 00:09:41,040 --> 00:09:45,470 that the twofold axis is a subgroup of the sixfold axis. 148 00:09:45,470 --> 00:09:49,160 And similarly, we can separate out three other elements, a 149 00:09:49,160 --> 00:09:59,470 onefold axis, same as A 2 pi; a 120-degree rotation, A 2 150 00:09:59,470 --> 00:10:08,360 pi/3; and a 240-degree rotation, A 4 pi/3. 151 00:10:08,360 --> 00:10:13,770 And these three operations, a group of rank 2, is what 152 00:10:13,770 --> 00:10:20,530 constitutes another subgroup, a subset of elements which by 153 00:10:20,530 --> 00:10:26,980 themselves satisfy all of the requirements of a group. 154 00:10:26,980 --> 00:10:32,360 So this is one example of how we can have more than one 155 00:10:32,360 --> 00:10:35,730 symmetry element, a set of operations existing about the 156 00:10:35,730 --> 00:10:37,550 same locus and space. 157 00:10:37,550 --> 00:10:44,640 This is not how one would go about the deriving these 158 00:10:44,640 --> 00:10:46,110 higher symmetry however. 159 00:10:46,110 --> 00:10:50,090 Because what we have to do is to add something to the set, 160 00:10:50,090 --> 00:10:53,680 something that's called an extender, and then show that 161 00:10:53,680 --> 00:10:58,150 all the requirements of a set being a group is satisfied; 162 00:10:58,150 --> 00:11:03,190 namely, that a combination of any two elements in the group 163 00:11:03,190 --> 00:11:07,030 is also a member the group, that for every operation an 164 00:11:07,030 --> 00:11:10,840 inverse exists, and the identity operation is a member 165 00:11:10,840 --> 00:11:11,570 of the group. 166 00:11:11,570 --> 00:11:14,670 So you can build up more higher symmetries, more 167 00:11:14,670 --> 00:11:18,270 complex symmetries by adding some operation called an 168 00:11:18,270 --> 00:11:21,680 extender and then taking all of the products of these 169 00:11:21,680 --> 00:11:27,120 elements and see what new operations arise. 170 00:11:27,120 --> 00:11:30,840 So we can have more than one symmetry element operating 171 00:11:30,840 --> 00:11:35,080 about the same locus and the set of individual operations 172 00:11:35,080 --> 00:11:37,480 that it embodies. 173 00:11:37,480 --> 00:11:42,970 And here sitting all by itself is a mirror plane. 174 00:11:42,970 --> 00:11:50,210 Why don't we combine a mirror plane with the rotation axis? 175 00:11:50,210 --> 00:11:52,210 And we've already seen examples of that. 176 00:11:52,210 --> 00:11:57,350 For example, in the square pattern that these tiles make 177 00:11:57,350 --> 00:12:00,220 up, there's a fourfold axis in the middle of each of the 178 00:12:00,220 --> 00:12:02,440 squares that are also mirror planes that pass 179 00:12:02,440 --> 00:12:04,200 through that location. 180 00:12:04,200 --> 00:12:06,570 So let me look at a combination that is a little 181 00:12:06,570 --> 00:12:08,410 simpler to handle. 182 00:12:08,410 --> 00:12:13,860 And what I'm going to say is here sits a mirror plane with 183 00:12:13,860 --> 00:12:17,300 a reflection operation that I'll call sigma 1 for the 184 00:12:17,300 --> 00:12:19,640 individual operation. 185 00:12:19,640 --> 00:12:23,790 And let me say that I combine now in that space a second 186 00:12:23,790 --> 00:12:27,390 reflection operation about a line that 187 00:12:27,390 --> 00:12:30,780 intersects the first one. 188 00:12:30,780 --> 00:12:34,610 And the question I'm going to ask now is happens when I take 189 00:12:34,610 --> 00:12:41,240 a first motif, number 1, reflected in the locus of the 190 00:12:41,240 --> 00:12:45,180 first mirror plane to get number 2, and then reflect 191 00:12:45,180 --> 00:12:49,455 that a second time in the locus of the second mirror 192 00:12:49,455 --> 00:12:52,100 plane to get one that sits up here. 193 00:12:54,800 --> 00:13:00,360 So the question is now what is the operation sigma 1 followed 194 00:13:00,360 --> 00:13:03,530 by the operation sigma 2 equals to? 195 00:13:08,260 --> 00:13:12,750 We can almost answer by the process of elimination. 196 00:13:12,750 --> 00:13:16,650 If this one is left handed, reflection changes the 197 00:13:16,650 --> 00:13:20,080 polarity, so number 2 is right hand. 198 00:13:20,080 --> 00:13:22,970 And if we reflect a second time, the right-handed one 199 00:13:22,970 --> 00:13:25,510 goes to a left-handed one. 200 00:13:25,510 --> 00:13:28,770 So what we're asking is how do I get from one left-handed 201 00:13:28,770 --> 00:13:33,210 motif to another left-handed motif? 202 00:13:33,210 --> 00:13:36,850 Of the three operations that exist in a two-dimensional 203 00:13:36,850 --> 00:13:39,640 space could be translation. 204 00:13:39,640 --> 00:13:42,860 But, clearly, I can't take the first one and slide it 205 00:13:42,860 --> 00:13:46,240 parallel to itself and make it coincide with the third. 206 00:13:46,240 --> 00:13:49,490 So translation would not change chirality, but that 207 00:13:49,490 --> 00:13:51,090 won't work. 208 00:13:51,090 --> 00:13:53,070 What's left? 209 00:13:53,070 --> 00:13:53,370 Rotation. 210 00:13:53,370 --> 00:13:54,310 Yes. 211 00:13:54,310 --> 00:13:56,110 You want to clarify something first? 212 00:13:56,110 --> 00:13:58,745 AUDIENCE: That third one, it looks kind of left-handed. 213 00:13:58,745 --> 00:13:59,020 [INAUDIBLE]. 214 00:13:59,020 --> 00:13:59,390 PROFESSOR: Oops. 215 00:13:59,390 --> 00:13:59,810 It does. 216 00:13:59,810 --> 00:14:00,050 It does indeed. 217 00:14:00,050 --> 00:14:01,050 Sorry about that. 218 00:14:01,050 --> 00:14:03,350 When you've got your nose poked right in these things, 219 00:14:03,350 --> 00:14:04,340 it's easy to overlook it. 220 00:14:04,340 --> 00:14:05,650 Yes. 221 00:14:05,650 --> 00:14:06,570 Absolutely right. 222 00:14:06,570 --> 00:14:08,390 That should be over canted over a bit more like I had it 223 00:14:08,390 --> 00:14:09,010 the first way. 224 00:14:09,010 --> 00:14:11,190 Very good thank you. 225 00:14:11,190 --> 00:14:12,270 I don't want to proceed much. 226 00:14:12,270 --> 00:14:12,580 Yes? 227 00:14:12,580 --> 00:14:15,710 AUDIENCE: What did you [INAUDIBLE] 228 00:14:15,710 --> 00:14:17,120 axis [INAUDIBLE]? 229 00:14:17,120 --> 00:14:19,220 PROFESSOR: We could do that. 230 00:14:19,220 --> 00:14:22,970 And let's do it, and we're going to encounter 231 00:14:22,970 --> 00:14:24,050 this sooner or later. 232 00:14:24,050 --> 00:14:30,240 So we have a first mirror plane reflect from a first 233 00:14:30,240 --> 00:14:34,050 one, which is right handed, to second one that's left handed. 234 00:14:34,050 --> 00:14:39,710 And then define a second locus, and that one would take 235 00:14:39,710 --> 00:14:43,020 the second one and move it over to here, the same 236 00:14:43,020 --> 00:14:45,970 distance on the other side of this mirror plane. 237 00:14:45,970 --> 00:14:51,430 And I'll let you answer the question since you asked it. 238 00:14:51,430 --> 00:14:53,720 How is the first one related to the third one? 239 00:14:53,720 --> 00:14:54,480 AUDIENCE: Translation. 240 00:14:54,480 --> 00:14:54,890 PROFESSOR: Translation. 241 00:14:54,890 --> 00:14:55,790 Yes. 242 00:14:55,790 --> 00:14:56,525 AUDIENCE: It's translated. 243 00:14:56,525 --> 00:15:03,010 PROFESSOR: Exactly So if this distance is delta, the first 244 00:15:03,010 --> 00:15:05,155 and the third are related be a translation 245 00:15:05,155 --> 00:15:07,800 which is twice delta. 246 00:15:07,800 --> 00:15:11,020 So that would be in addition where we'll eventually want to 247 00:15:11,020 --> 00:15:14,200 consider when we have patterns that are based on a lattice. 248 00:15:14,200 --> 00:15:19,160 But right now, I want to consider symmetry operations 249 00:15:19,160 --> 00:15:22,940 that leave at least a point in space and move perhaps a line 250 00:15:22,940 --> 00:15:25,730 in the case of the mirror plane. 251 00:15:25,730 --> 00:15:29,130 But back to the original question, what is this first 252 00:15:29,130 --> 00:15:33,020 reflection followed by a second reflection? 253 00:15:33,020 --> 00:15:34,610 We've just thrown out translation. 254 00:15:34,610 --> 00:15:36,970 That won't work unless the mirror planes are parallel to 255 00:15:36,970 --> 00:15:37,460 one of those. 256 00:15:37,460 --> 00:15:40,220 The only thing left is rotation. 257 00:15:40,220 --> 00:15:44,380 And in point of fact, even looking at this diagram, you 258 00:15:44,380 --> 00:15:50,010 can get from the first one to the third one by a rotation. 259 00:15:50,010 --> 00:15:53,930 So the answer to the question is that a combination of two 260 00:15:53,930 --> 00:15:59,200 reflection operations is a rotation operation about the 261 00:15:59,200 --> 00:16:02,740 point of intersection of the mirror planes. 262 00:16:02,740 --> 00:16:05,940 But we can go further and say exactly how large this 263 00:16:05,940 --> 00:16:07,440 rotation is. 264 00:16:07,440 --> 00:16:09,580 Let's draw an extra line in here. 265 00:16:09,580 --> 00:16:12,610 And let's say that this angle is alpha. 266 00:16:12,610 --> 00:16:14,790 And if I repeat this by reflection, that 267 00:16:14,790 --> 00:16:18,410 angle is also alpha. 268 00:16:18,410 --> 00:16:20,860 Let's label this angle beta. 269 00:16:20,860 --> 00:16:24,630 Repeat that by reflection, and that angle must also be beta. 270 00:16:24,630 --> 00:16:31,360 The angle between the two mirror planes is the angle mu. 271 00:16:31,360 --> 00:16:36,860 And clearly mu is equal to alpha plus beta. 272 00:16:36,860 --> 00:16:41,130 So the answer to the question quite generally is have two 273 00:16:41,130 --> 00:16:45,410 mirror planes intersect at an angle mu, and their successive 274 00:16:45,410 --> 00:16:49,200 operation is going to be equivalent to the net result 275 00:16:49,200 --> 00:16:52,050 of rotating through twice of the angle 276 00:16:52,050 --> 00:16:55,580 between them mirror line. 277 00:16:55,580 --> 00:16:57,600 So we don't have to think about that anymore now. 278 00:16:57,600 --> 00:16:59,250 We know if we combine to mirror planes 279 00:16:59,250 --> 00:17:00,500 we get a net rotation. 280 00:17:03,980 --> 00:17:09,420 So that is a basic result, and that's a first example of 281 00:17:09,420 --> 00:17:12,160 something that I'm going to call by my own pet term. 282 00:17:12,160 --> 00:17:14,859 I'm going to call them combination theorem, the 283 00:17:14,859 --> 00:17:18,190 theorem that tells you what you get when you combine two 284 00:17:18,190 --> 00:17:19,380 operations. 285 00:17:19,380 --> 00:17:23,079 And what this is going to give us is a way of filling in one 286 00:17:23,079 --> 00:17:25,160 box in the group multiplication table. 287 00:17:28,730 --> 00:17:35,860 So knowing what two mirror planes combined at an angle 288 00:17:35,860 --> 00:17:42,500 should be, I can now combine the mirror planes at angles 289 00:17:42,500 --> 00:17:45,660 which gives me a rotation operation which is one of the 290 00:17:45,660 --> 00:17:49,010 ones that's allowed for a lattice. 291 00:17:49,010 --> 00:17:51,500 In general, if I'm not yet putting these symmetry 292 00:17:51,500 --> 00:17:54,850 elements in a lattice, I could combine them in such a way 293 00:17:54,850 --> 00:17:58,940 such that I got a 17-fold rotation as the angle. 294 00:17:58,940 --> 00:18:01,540 And then the angle between the mirror planes would 295 00:18:01,540 --> 00:18:04,810 be half of 2 pi/17. 296 00:18:04,810 --> 00:18:08,170 Could be a lovely symmetry, nice for a pendant or a ring, 297 00:18:08,170 --> 00:18:10,820 but not for a lattice and not for a crystal. 298 00:18:10,820 --> 00:18:13,130 So I'm going to want to combine mirror planes at 299 00:18:13,130 --> 00:18:19,040 angles that correspond to rotations that are compatible 300 00:18:19,040 --> 00:18:21,100 with the lattice. 301 00:18:21,100 --> 00:18:24,660 So what I could do is take a first reflection 302 00:18:24,660 --> 00:18:26,695 locus, sigma 1. 303 00:18:26,695 --> 00:18:32,280 And if I want the rotation angle to be 180 degrees for a 304 00:18:32,280 --> 00:18:38,550 twofold axis, I would want the second mirror plane with the 305 00:18:38,550 --> 00:18:45,100 operation sigma 2 to be at one half of pi with 306 00:18:45,100 --> 00:18:47,230 respect to the first. 307 00:18:47,230 --> 00:18:52,190 And for this specific combination then, reflection 308 00:18:52,190 --> 00:18:59,130 in the first mirror planes followed by reflection in the 309 00:18:59,130 --> 00:19:04,480 second mirror planes to give me a third one, is going to be 310 00:19:04,480 --> 00:19:05,500 equivalent to-- 311 00:19:05,500 --> 00:19:07,160 Oops. 312 00:19:07,160 --> 00:19:08,390 Back here-- 313 00:19:08,390 --> 00:19:11,065 it's going to be equivalent to a twofold axis. 314 00:19:15,010 --> 00:19:22,460 So sigma 1 followed by sigma 2 at an angle one half of pi is 315 00:19:22,460 --> 00:19:25,010 going to be equivalent to the rotation operation A pi. 316 00:19:38,850 --> 00:19:42,990 This is not yet a complete pattern because here's a 317 00:19:42,990 --> 00:19:46,395 mirror planes that wants to reflect this motifs as well. 318 00:19:49,300 --> 00:19:55,500 So let's just reflect number 3 across the mirror planes 1. 319 00:19:55,500 --> 00:19:58,390 And then we're going to get a fourth object. 320 00:19:58,390 --> 00:20:01,880 Now let's show that this constitutes a group. 321 00:20:01,880 --> 00:20:04,100 We've got the identity operation doing nothing. 322 00:20:09,850 --> 00:20:12,490 We've got the operation sigma 1. 323 00:20:12,490 --> 00:20:15,730 We've got the operation sigma 2, and we've got 324 00:20:15,730 --> 00:20:18,820 the operation a pi. 325 00:20:18,820 --> 00:20:23,380 I'll put the same operations again in the vertical column. 326 00:20:23,380 --> 00:20:27,310 Here's operation 1 doing nothing, sigma 1, 327 00:20:27,310 --> 00:20:29,360 sigma 2, and A pi. 328 00:20:32,050 --> 00:20:36,090 So doing nothing and following it by one of these four 329 00:20:36,090 --> 00:20:40,600 operations just gives you the same thing back again. 330 00:20:40,600 --> 00:20:44,790 And doing these operations followed by 1 will give me the 331 00:20:44,790 --> 00:20:46,535 same operation back again. 332 00:20:49,300 --> 00:20:54,940 If I reflect twice across sigma 1 from left to right and 333 00:20:54,940 --> 00:20:57,690 right to left, I'm back to where I started from. 334 00:20:57,690 --> 00:21:03,250 So sigma 1 followed by sigma 1 is the identity operation. 335 00:21:03,250 --> 00:21:08,150 Sigma 1 followed by sigma 2 is what we just did. 336 00:21:08,150 --> 00:21:11,520 That turns out to be the operation A pi. 337 00:21:11,520 --> 00:21:16,800 And sigma 1 reflection and following that by A pi is the 338 00:21:16,800 --> 00:21:18,890 way in which I get to the fourth one. 339 00:21:18,890 --> 00:21:25,210 So that is going to be equal to sigma 2. 340 00:21:25,210 --> 00:21:27,320 Well, let me rips through the last ones fairly quickly. 341 00:21:27,320 --> 00:21:31,060 Sigma 2 followed by signal 1. 342 00:21:31,060 --> 00:21:34,720 Sigma 2 followed by sigma 1 is the same as A pi. 343 00:21:37,620 --> 00:21:40,590 Sigma 2 followed by sigma 2 gets us back to where we 344 00:21:40,590 --> 00:21:41,220 started from. 345 00:21:41,220 --> 00:21:43,400 That's the identity operation. 346 00:21:43,400 --> 00:21:48,290 Sigma 2 followed by A pi reflect from here to here and 347 00:21:48,290 --> 00:21:49,290 then rotate. 348 00:21:49,290 --> 00:21:53,150 That's the same as sigma 1. 349 00:21:53,150 --> 00:21:57,090 And the final sequence, rotation followed by 350 00:21:57,090 --> 00:21:59,930 reflection would give me sigma 2. 351 00:21:59,930 --> 00:22:03,430 Rotation followed by sigma 2 will give me sigma 1. 352 00:22:03,430 --> 00:22:07,120 And doing the rotation operation twice is the same as 353 00:22:07,120 --> 00:22:09,890 the identity operation. 354 00:22:09,890 --> 00:22:11,823 So are the group postulates satisfied? 355 00:22:14,700 --> 00:22:16,270 Yes. 356 00:22:16,270 --> 00:22:25,190 The combination of any two elements is a 357 00:22:25,190 --> 00:22:26,440 member of the group. 358 00:22:37,840 --> 00:22:40,813 For every operation, an inverse exists. 359 00:22:50,000 --> 00:22:55,670 And we can answer that question very easily by merely 360 00:22:55,670 --> 00:22:59,950 looking at the column under a particular element, and 361 00:22:59,950 --> 00:23:03,700 somewhere we find the identity operation. 362 00:23:03,700 --> 00:23:08,090 So sigma 1 is its own inverse. 363 00:23:08,090 --> 00:23:11,690 So, yes, an inverse exists for every operation. 364 00:23:11,690 --> 00:23:14,715 And then finally, identity is a member of the group. 365 00:23:20,240 --> 00:23:22,430 So everything's lights up. 366 00:23:22,430 --> 00:23:24,260 Bells ring. 367 00:23:24,260 --> 00:23:26,940 This set of four elements is entitled to 368 00:23:26,940 --> 00:23:30,680 call itself a group. 369 00:23:30,680 --> 00:23:36,490 And in particular, it is a group of rank 4 because there 370 00:23:36,490 --> 00:23:39,370 are four elements in the group. 371 00:23:39,370 --> 00:23:40,900 I show you a cool thing. 372 00:23:40,900 --> 00:23:45,560 I noticed that the number of objects in the pattern, the 373 00:23:45,560 --> 00:23:48,550 number of motifs in the pattern, is exactly the same 374 00:23:48,550 --> 00:23:50,480 as the order of the group. 375 00:23:50,480 --> 00:23:51,520 Why? 376 00:23:51,520 --> 00:23:56,390 Because these four operations tell you how to get from any 377 00:23:56,390 --> 00:24:01,500 operation to the remaining three and tell you how to get 378 00:24:01,500 --> 00:24:02,260 it into itself. 379 00:24:02,260 --> 00:24:04,290 So identity relates it to itself. 380 00:24:04,290 --> 00:24:06,080 Sigma 1 reflects it to here. 381 00:24:06,080 --> 00:24:08,190 A pi rotates it down to here. 382 00:24:08,190 --> 00:24:10,740 Sigma 2 reflects it down to here. 383 00:24:10,740 --> 00:24:14,300 So there's always a one-to-one correspondence between the 384 00:24:14,300 --> 00:24:17,770 elements that are in the group and the rank of a group and 385 00:24:17,770 --> 00:24:21,050 the number of objects that are in the pattern. 386 00:24:21,050 --> 00:24:21,380 Yes, sir? 387 00:24:21,380 --> 00:24:23,970 AUDIENCE: How do you show if the inverse exists again? 388 00:24:23,970 --> 00:24:29,660 PROFESSOR: The inverse exists if I can find something that 389 00:24:29,660 --> 00:24:34,170 combined with an element in my basic set gives me the 390 00:24:34,170 --> 00:24:36,192 identity operation. 391 00:24:36,192 --> 00:24:39,860 So it's the operation followed by its inverse has to be the 392 00:24:39,860 --> 00:24:41,630 same as doing nothing. 393 00:24:41,630 --> 00:24:44,530 So if I look under each element and find the identity 394 00:24:44,530 --> 00:24:48,655 operation, then sigma 2 is its own inverse. 395 00:24:48,655 --> 00:24:52,680 If I look under A pi, A pi is its own inverse. 396 00:24:52,680 --> 00:24:53,970 This is generally not the case. 397 00:24:53,970 --> 00:24:55,535 This is a very simple symmetry. 398 00:25:04,420 --> 00:25:07,250 So really we can work with commas and little figures, and 399 00:25:07,250 --> 00:25:08,970 that's one way of getting these symmetries. 400 00:25:08,970 --> 00:25:12,050 But group theory I think you can see is a very nice, very 401 00:25:12,050 --> 00:25:15,840 elegant language for describing some 402 00:25:15,840 --> 00:25:17,135 characteristics of these patterns. 403 00:25:19,640 --> 00:25:20,890 Now little bit of jargon. 404 00:25:23,485 --> 00:25:27,370 As if this material were not confusing enough, there are 405 00:25:27,370 --> 00:25:32,780 two different languages that are used to denote these 406 00:25:32,780 --> 00:25:35,560 unique combinations. 407 00:25:35,560 --> 00:25:40,530 This notation where we simply have a number for a rotation 408 00:25:40,530 --> 00:25:45,310 axis or an m for mirror plane is something that was first 409 00:25:45,310 --> 00:25:49,290 proposed by two 410 00:25:49,290 --> 00:25:52,170 mathematicians, Hermann and Mauguin. 411 00:25:55,830 --> 00:26:03,360 And this was adopted as the preferred notation in the 412 00:26:03,360 --> 00:26:07,660 international tables, that hefty tome that I brought it 413 00:26:07,660 --> 00:26:08,715 on the first day of classes. 414 00:26:08,715 --> 00:26:12,410 So this is also referred to synonymously as the 415 00:26:12,410 --> 00:26:13,660 international notation. 416 00:26:21,100 --> 00:26:24,370 Let me put these things out horizontally. 417 00:26:24,370 --> 00:26:30,930 For rotation axes by themselves, the symbol in 418 00:26:30,930 --> 00:26:34,280 Hermann notation is simply n. 419 00:26:34,280 --> 00:26:37,710 And for lattice, n-- 420 00:26:37,710 --> 00:26:40,030 as we've seen many times-- 421 00:26:40,030 --> 00:26:45,090 is restricted to what in Hermann-Mauguin notation would 422 00:26:45,090 --> 00:26:49,475 be either 1, 2, 3, 4, and 6. 423 00:26:52,820 --> 00:26:57,670 There's another type of notation that is use quite 424 00:26:57,670 --> 00:27:03,630 frequently by condensed matter physicist, and this is called 425 00:27:03,630 --> 00:27:12,380 the Schoenflies notation after one of the mathematicians who 426 00:27:12,380 --> 00:27:20,660 was the first to derive the three-dimensional symmetries. 427 00:27:20,660 --> 00:27:22,130 This was a curious thing. 428 00:27:22,130 --> 00:27:25,530 There were three people in different parts of the world 429 00:27:25,530 --> 00:27:27,830 about the turn of the century-- and this was before 430 00:27:27,830 --> 00:27:30,170 email or even air mail-- 431 00:27:30,170 --> 00:27:33,520 who were trying to derive the three-dimensional symmetries. 432 00:27:33,520 --> 00:27:36,450 Schoenflies was one of them. 433 00:27:36,450 --> 00:27:40,430 And all three of them got to the same place at about the 434 00:27:40,430 --> 00:27:45,220 same time, and the results were obtained. 435 00:27:45,220 --> 00:27:47,090 But each one used his own method and had 436 00:27:47,090 --> 00:27:48,350 his different notation. 437 00:27:48,350 --> 00:27:51,470 Schoenflies was one of the people who did this. 438 00:27:51,470 --> 00:27:58,190 Schoenflies was a mathematician and based his 439 00:27:58,190 --> 00:27:59,990 notation on group theory. 440 00:28:02,530 --> 00:28:05,190 Something called a cyclic group is what 441 00:28:05,190 --> 00:28:06,610 the C stands for. 442 00:28:12,010 --> 00:28:16,170 And a cyclic group is one in which all elements are power, 443 00:28:16,170 --> 00:28:29,660 so to speak, of some basic operation. 444 00:28:39,720 --> 00:28:43,390 For example, a fourfold axis consists of the set of 445 00:28:43,390 --> 00:28:50,010 operations A pi/2; a 90 degree rotation; A pi, which can be 446 00:28:50,010 --> 00:28:55,160 written as A pi over 2 squared; A pi over 2 squared; 447 00:28:55,160 --> 00:28:55,750 [? then we need ?] 448 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 449 00:29:04,470 --> 00:29:08,660 doing A pi/2 three times; and finally, the identity 450 00:29:08,660 --> 00:29:13,450 operation, which is the same as A 2 pi, which is equivalent 451 00:29:13,450 --> 00:29:18,160 to doing the basic operation A pi/2 four times. 452 00:29:18,160 --> 00:29:21,830 So the group of rank 4, there are four elements in the 453 00:29:21,830 --> 00:29:26,790 group, each one is a power of the basic operation A pi/2. 454 00:29:26,790 --> 00:29:30,840 So a rotation axis by itself as a cyclic group. 455 00:29:30,840 --> 00:29:34,740 Schoenflies indicates these generically by C subscript n, 456 00:29:34,740 --> 00:29:36,890 where n is the rank of the axis. 457 00:29:36,890 --> 00:29:38,870 Hence, this is C1. 458 00:29:38,870 --> 00:29:40,540 This is C2. 459 00:29:40,540 --> 00:29:42,230 This is C3. 460 00:29:42,230 --> 00:29:43,400 This is C4. 461 00:29:43,400 --> 00:29:44,650 And this is C6. 462 00:29:54,220 --> 00:29:57,670 A mirror plane by itself is indicated m in the 463 00:29:57,670 --> 00:30:02,300 international notation. 464 00:30:02,300 --> 00:30:04,810 A mirror plane is also a cyclic group. 465 00:30:04,810 --> 00:30:08,240 There are two operations, reflection and reflection back 466 00:30:08,240 --> 00:30:09,290 to where you came from. 467 00:30:09,290 --> 00:30:11,730 Doing the reflection operation twice is the 468 00:30:11,730 --> 00:30:13,160 same as doing nothing. 469 00:30:13,160 --> 00:30:18,990 So Schoenflies also call this one C. And the one thing that 470 00:30:18,990 --> 00:30:27,520 is without meaning in English is his subscript S. And 471 00:30:27,520 --> 00:30:33,880 Schoenflies was German, and the S stands for spiegel, 472 00:30:33,880 --> 00:30:37,020 which is the German word for mirror. 473 00:30:37,020 --> 00:30:39,170 We have in many cities a paper that's 474 00:30:39,170 --> 00:30:40,770 called the Daily Mirror. 475 00:30:40,770 --> 00:30:44,252 In Germany, there's a paper called Der Spiegel. 476 00:30:44,252 --> 00:30:46,950 So they use it the same way in everyday life 477 00:30:46,950 --> 00:30:48,620 as well as in notation. 478 00:30:48,620 --> 00:30:51,840 So C sub S in the Schoenflies notation is a mirror plane. 479 00:30:51,840 --> 00:30:55,340 Cyclic group the S stand for spiegel. 480 00:30:55,340 --> 00:31:07,990 For the combinations of which we have seen only one, in the 481 00:31:07,990 --> 00:31:12,560 Schoenflies notation these are all of the form Cnv, a 482 00:31:12,560 --> 00:31:16,180 rotation axis Cn with a vertical mirror plane passing 483 00:31:16,180 --> 00:31:16,940 through it. 484 00:31:16,940 --> 00:31:19,310 So this one, for example-- we've yet 485 00:31:19,310 --> 00:31:21,000 to look at the others-- 486 00:31:21,000 --> 00:31:23,620 would be C2v. 487 00:31:23,620 --> 00:31:26,170 So we'll have to draw the first, and then I'll give you 488 00:31:26,170 --> 00:31:28,630 the complete set. 489 00:31:28,630 --> 00:31:32,410 So this is a rotation axis with a mirror plane passing 490 00:31:32,410 --> 00:31:35,170 vertically through it. 491 00:31:35,170 --> 00:31:36,450 Let's do a few more. 492 00:31:36,450 --> 00:31:42,210 And I think having done a couple in great detail, we'll 493 00:31:42,210 --> 00:31:46,040 see what the others will look like quite readily. 494 00:31:46,040 --> 00:31:51,630 Let's take a fourfold axis and add to the rotation operations 495 00:31:51,630 --> 00:32:00,870 A pi/2, a first mirror plane that I'll label sigma 1. 496 00:32:00,870 --> 00:32:04,630 I can permute the order of operations and say that sigma 497 00:32:04,630 --> 00:32:09,430 1 followed by the rotation operation A pi/2 is going to 498 00:32:09,430 --> 00:32:14,120 be equal to a second reflection operation that is 499 00:32:14,120 --> 00:32:20,830 equal to one half of pi/2 away from the first. 500 00:32:20,830 --> 00:32:26,590 So I'm claiming that if I reflect in the first mirror 501 00:32:26,590 --> 00:32:31,610 plane and then rotate by 90 degrees that should be another 502 00:32:31,610 --> 00:32:35,010 mirror plane at 45 degrees to the first. 503 00:32:35,010 --> 00:32:38,110 So let me do exactly what I advertised. 504 00:32:38,110 --> 00:32:40,150 Here's the first, one right handed. 505 00:32:40,150 --> 00:32:43,640 Reflect across to get a second one, which is left handed. 506 00:32:43,640 --> 00:32:50,820 Reflect, then rotate, and that would bring my left-handed one 507 00:32:50,820 --> 00:32:52,610 up to this location here. 508 00:32:52,610 --> 00:32:54,860 So here's 3, and it's left handed. 509 00:32:54,860 --> 00:32:57,700 And lo and behold, just as advertised, I get from the 510 00:32:57,700 --> 00:33:00,720 first one to the third one by reflecting across a mirror 511 00:33:00,720 --> 00:33:05,020 plane, which is one half of pi/2. 512 00:33:09,720 --> 00:33:13,890 If let these symmetry elements operate on each other, what 513 00:33:13,890 --> 00:33:15,990 I'll end up with is a set of mirror planes 514 00:33:15,990 --> 00:33:19,090 at 45 degree intervals. 515 00:33:19,090 --> 00:33:25,960 And what I'll have is a pair of objects hung in the same 516 00:33:25,960 --> 00:33:28,625 fashion at every mirror plane. 517 00:33:33,810 --> 00:33:35,550 Is that too fast? 518 00:33:35,550 --> 00:33:36,870 You want to go through that a little slower? 519 00:33:40,390 --> 00:33:42,160 Oh, go ahead say, go through it more slowly. 520 00:33:42,160 --> 00:33:44,810 Everybody's afraid to say, yeah, yeah, and been seem like 521 00:33:44,810 --> 00:33:45,620 a class dummy. 522 00:33:45,620 --> 00:33:46,870 Do you want me to do it more slowly? 523 00:33:51,090 --> 00:33:54,430 I see people still writing, so I think what you'd rather have 524 00:33:54,430 --> 00:33:57,200 is me be quiet for a bit while you catch up to where I am. 525 00:34:06,900 --> 00:34:14,679 In the Schoenflies notation, this lovely thing here is C4. 526 00:34:14,679 --> 00:34:19,860 It's a rotation axis of rank 4 with a vertical mirror plane 527 00:34:19,860 --> 00:34:21,860 added to it. 528 00:34:21,860 --> 00:34:25,290 And the international notation, now the 529 00:34:25,290 --> 00:34:28,929 Hermann-Mauguin notation, is just a running list of the 530 00:34:28,929 --> 00:34:32,830 individual symmetry elements that are present. 531 00:34:32,830 --> 00:34:37,770 And now I really are going to have a mouthful. 532 00:34:37,770 --> 00:34:41,040 This is a fourfold axis, so the symbol for that is 4. 533 00:34:41,040 --> 00:34:42,150 This is a mirror plane. 534 00:34:42,150 --> 00:34:43,239 This is a mirror plane. 535 00:34:43,239 --> 00:34:44,260 This is a mirror plane. 536 00:34:44,260 --> 00:34:45,190 This is a mirror plane. 537 00:34:45,190 --> 00:34:46,515 This is a mirror plane. 538 00:34:46,515 --> 00:34:48,199 This is a mirror plane. 539 00:34:48,199 --> 00:34:49,820 That's a mirror plane, and that's a mirror plane. 540 00:34:49,820 --> 00:34:58,750 So it looks as though I should call this 4mmmmmmmm, which 541 00:34:58,750 --> 00:35:00,000 comes 4mmmmmmmm. 542 00:35:02,130 --> 00:35:04,460 Well, that is a typical reaction. 543 00:35:04,460 --> 00:35:06,880 It's a lovely symmetry. 544 00:35:06,880 --> 00:35:10,120 But that is a mouthful. 545 00:35:10,120 --> 00:35:14,630 So it isn't really necessary to give all these m's So the 546 00:35:14,630 --> 00:35:24,370 international notation is a running list of the 547 00:35:24,370 --> 00:35:28,210 independent symmetry elements, and that's the new wrinkle 548 00:35:28,210 --> 00:35:31,110 that I'm introducing. 549 00:35:31,110 --> 00:35:34,305 It's a running list of the independent symmetry elements. 550 00:35:41,254 --> 00:35:46,570 And all these mirror planes are just different sigma that 551 00:35:46,570 --> 00:35:49,090 exist as operations in the group. 552 00:35:49,090 --> 00:35:52,390 But how many different kinds of mirror planes are they? 553 00:35:52,390 --> 00:35:57,060 Well, there are two different kinds of mirror planes, both 554 00:35:57,060 --> 00:36:01,630 in terms of the way in which they function in the pattern. 555 00:36:01,630 --> 00:36:05,380 The two motifs related by reflection hang close to this 556 00:36:05,380 --> 00:36:09,550 mirror plane, but they're widely separated for this 557 00:36:09,550 --> 00:36:10,850 mirror plane. 558 00:36:10,850 --> 00:36:15,100 So the motifs do different things relative to those two 559 00:36:15,100 --> 00:36:16,630 kinds of mirror planes. 560 00:36:16,630 --> 00:36:20,420 Another way of asking what's independent is if I start with 561 00:36:20,420 --> 00:36:25,210 this 1 mirror plane and repeat it by 90-degree rotations, 562 00:36:25,210 --> 00:36:29,660 I'll get these 4 mirror planes 90 degrees away. 563 00:36:29,660 --> 00:36:34,950 So they are not independent in that these mirror planes are 564 00:36:34,950 --> 00:36:39,140 all related by the rotational symmetry that's present. 565 00:36:39,140 --> 00:36:42,630 You don't get this mirror plane in any fashion other 566 00:36:42,630 --> 00:36:46,730 than saying, if I combine the rotation operation with this 567 00:36:46,730 --> 00:36:50,970 reflection operation sigma, the net result is this 568 00:36:50,970 --> 00:36:52,230 reflection plane. 569 00:36:52,230 --> 00:36:55,310 So there are two mirror planes that are distinct in this 570 00:36:55,310 --> 00:36:57,440 arrangement of symmetry elements, distinct in the 571 00:36:57,440 --> 00:37:01,550 sense that they function in different ways in the pattern; 572 00:37:01,550 --> 00:37:05,380 distinct in the sense that no other operation that is 573 00:37:05,380 --> 00:37:10,160 present will throw these two operations into one another. 574 00:37:10,160 --> 00:37:15,660 Another example, and this is in fact symmetry 4mm, is the 575 00:37:15,660 --> 00:37:17,300 square tile. 576 00:37:17,300 --> 00:37:19,670 If you look at the mirror planes there, they are 45 577 00:37:19,670 --> 00:37:20,730 degrees apart. 578 00:37:20,730 --> 00:37:23,850 But one of those mirror planes comes out normal to the edge 579 00:37:23,850 --> 00:37:25,060 of the tile. 580 00:37:25,060 --> 00:37:28,710 The other mirror plane comes out of the vertex of the tile. 581 00:37:28,710 --> 00:37:32,390 So they are different in the way they function in the space 582 00:37:32,390 --> 00:37:34,450 which has this symmetry. 583 00:37:34,450 --> 00:37:38,110 So, mercifully, we don't have to call this 4mmmmmmmm. 584 00:37:38,110 --> 00:37:42,750 We drop the last 6 m's, and this one is called simply for 585 00:37:42,750 --> 00:37:47,360 4mm, 2 kinds of mirror planes with a fourfold axis. 586 00:37:47,360 --> 00:37:49,990 If you're familiar with the way in which these were 587 00:37:49,990 --> 00:37:55,780 derived, the symbol tells you exactly what you've got. 588 00:37:55,780 --> 00:38:00,510 So it's a very useful notation for these symmetries. 589 00:38:03,770 --> 00:38:06,820 Let me pause here, give you a chance to catch up. 590 00:38:06,820 --> 00:38:07,302 Yes, sir? 591 00:38:07,302 --> 00:38:09,471 AUDIENCE: So wait, all you did here is take two separate 592 00:38:09,471 --> 00:38:12,122 mirror planes at an angle of 45 degrees between, and you 593 00:38:12,122 --> 00:38:13,770 ended up with a fourfold symmetry. 594 00:38:13,770 --> 00:38:14,030 PROFESSOR: Exactly. 595 00:38:14,030 --> 00:38:17,564 AUDIENCE: But you didn't put on the fourfold symmetry, it 596 00:38:17,564 --> 00:38:19,060 was just what fell out of it? 597 00:38:19,060 --> 00:38:20,545 PROFESSOR: No. 598 00:38:20,545 --> 00:38:23,730 I Got this mirror plane to begin with by combining this 599 00:38:23,730 --> 00:38:27,770 reflection operation sigma with the operation A pi/2, and 600 00:38:27,770 --> 00:38:29,340 that's where this operation came from. 601 00:38:34,440 --> 00:38:37,480 But a general theorem that says if I have a rotation 602 00:38:37,480 --> 00:38:43,020 operation A alpha and combine it with a reflection operation 603 00:38:43,020 --> 00:38:48,640 sigma, the combined effect is a reflection operation sigma 604 00:38:48,640 --> 00:38:54,510 2, which is alpha/2 away from the first. 605 00:38:54,510 --> 00:38:57,360 So, again, showing you for this now rather messy diagram, 606 00:38:57,360 --> 00:39:01,290 if I start with the reflection operation that takes 1 of them 607 00:39:01,290 --> 00:39:05,890 throws of the 2 and then rotates 2 up to location 3, 608 00:39:05,890 --> 00:39:12,510 the way I get from 1 to 3 in one shot is to reflect in this 609 00:39:12,510 --> 00:39:13,760 locus sigma 2. 610 00:39:16,420 --> 00:39:16,850 Yes, sir? 611 00:39:16,850 --> 00:39:18,188 AUDIENCE: You could have started with any one of those 612 00:39:18,188 --> 00:39:20,232 two and got the third. 613 00:39:20,232 --> 00:39:21,690 PROFESSOR: Absolutely. 614 00:39:21,690 --> 00:39:23,250 Absolutely. 615 00:39:23,250 --> 00:39:27,370 The operations that are present operate 616 00:39:27,370 --> 00:39:30,090 everything in the space. 617 00:39:30,090 --> 00:39:33,460 So when I say that there's a mirror plane here that relates 618 00:39:33,460 --> 00:39:36,340 this one to this one, it also relates this one to this one, 619 00:39:36,340 --> 00:39:38,845 this one to this one, this one to this one, that mirror 620 00:39:38,845 --> 00:39:40,720 planes operates on everything. 621 00:39:40,720 --> 00:39:44,070 You can't say that a symmetry operation grab this little 622 00:39:44,070 --> 00:39:47,060 packet of space and moves it to another packet 623 00:39:47,060 --> 00:39:48,360 removed from it. 624 00:39:48,360 --> 00:39:52,000 To say it's a symmetry of a pattern or of a crystal, it 625 00:39:52,000 --> 00:39:54,850 has to leave everything invariant. 626 00:39:54,850 --> 00:39:56,100 OK. 627 00:39:59,200 --> 00:40:02,630 But, again, in terms of the language of groups theory, if 628 00:40:02,630 --> 00:40:06,730 you combine this pair of operations, then when you 629 00:40:06,730 --> 00:40:10,560 combine everything that you get pairwise, you will get in 630 00:40:10,560 --> 00:40:15,720 this case a total of, how many operations? 631 00:40:15,720 --> 00:40:19,390 How many operations are in this pattern? 632 00:40:19,390 --> 00:40:22,570 I said a moment ago the rank of the group is the number of 633 00:40:22,570 --> 00:40:24,130 operations that are present. 634 00:40:24,130 --> 00:40:26,860 It's the number of objects that are in the group because 635 00:40:26,860 --> 00:40:29,040 each operation in a group tells you how to get from 636 00:40:29,040 --> 00:40:30,520 anyone to all of the other. 637 00:40:30,520 --> 00:40:34,336 So, 2, 4, 6, 8, this is a group of rank 8, and there are 638 00:40:34,336 --> 00:40:36,911 8 operation that are present. 639 00:40:36,911 --> 00:40:39,320 And I can rattle them off quickly. 640 00:40:39,320 --> 00:40:43,390 Four operations, identity, 90, 180, 270 for 641 00:40:43,390 --> 00:40:44,340 the fourfold axis. 642 00:40:44,340 --> 00:40:46,770 That's 4. 643 00:40:46,770 --> 00:40:48,230 This mirror plane. 644 00:40:48,230 --> 00:40:50,230 This mirror plane. 645 00:40:50,230 --> 00:40:51,550 The mirror plane. 646 00:40:51,550 --> 00:40:52,470 And this mirror plane. 647 00:40:52,470 --> 00:40:54,130 That's 8. 648 00:40:54,130 --> 00:40:55,950 Four for rotation. 649 00:40:55,950 --> 00:40:57,200 Four for reflection. 650 00:41:06,370 --> 00:41:07,620 Other questions? 651 00:41:13,205 --> 00:41:13,860 All right. 652 00:41:13,860 --> 00:41:18,130 Let me then wrap up this quickly and get 653 00:41:18,130 --> 00:41:19,040 onto something new. 654 00:41:19,040 --> 00:41:21,715 Did I hear the hiccup or a question? 655 00:41:21,715 --> 00:41:22,100 No. 656 00:41:22,100 --> 00:41:25,200 It was a hiccup. 657 00:41:25,200 --> 00:41:26,450 Not a yawn I hope. 658 00:41:38,630 --> 00:41:42,830 Let me do the highest symmetry of all in two dimensions, and 659 00:41:42,830 --> 00:41:47,220 this is if I take a sixfold axis and combine it with a 660 00:41:47,220 --> 00:41:48,630 mirror plane. 661 00:41:48,630 --> 00:41:52,030 So here's the first operation sigma. 662 00:41:52,030 --> 00:41:55,190 This one is something of a bear to draw. 663 00:41:55,190 --> 00:41:56,700 This is sigma 1. 664 00:41:56,700 --> 00:42:02,050 Sigma 1 followed by the operation A 2 pi/6 should be 665 00:42:02,050 --> 00:42:09,840 equal to a reflection that is an angle one half of 2 pi/6 30 666 00:42:09,840 --> 00:42:11,170 degrees away from the first. 667 00:42:11,170 --> 00:42:14,250 So this will be sigma 2. 668 00:42:14,250 --> 00:42:21,530 So if I reflect from 1 to 2 and then rotate by 60 degrees, 669 00:42:21,530 --> 00:42:24,755 I'm going to get one that sets up here. 670 00:42:24,755 --> 00:42:29,760 And the way I get from number 1 to number 3 in one shot is 671 00:42:29,760 --> 00:42:31,570 by a reflection sigma 2. 672 00:42:31,570 --> 00:42:34,790 Now if I draw in all of those are mirror planes when they 673 00:42:34,790 --> 00:42:41,220 are repeated by the sixfold axis, I shouldn't have given 674 00:42:41,220 --> 00:42:43,900 this my 6mmmmmm treatment because there 675 00:42:43,900 --> 00:42:45,710 are 12 mirror planes. 676 00:42:45,710 --> 00:42:49,990 Two objects hanging on this one in one fashion spaced in a 677 00:42:49,990 --> 00:42:52,270 different way that are belly to belly. 678 00:42:52,270 --> 00:42:55,030 They're back to back here, so it does something different in 679 00:42:55,030 --> 00:42:56,040 the pattern. 680 00:42:56,040 --> 00:42:57,430 A pair hanging here. 681 00:42:57,430 --> 00:43:01,920 A pair hanging in here. 682 00:43:01,920 --> 00:43:05,262 A pair hanging and here. 683 00:43:05,262 --> 00:43:06,920 A pair hanging in here. 684 00:43:09,910 --> 00:43:13,770 And finally, a pair hanging in here. 685 00:43:13,770 --> 00:43:17,730 So there are a total 2, 4, 6, 8, 10, 12 686 00:43:17,730 --> 00:43:19,350 motifs in this pattern. 687 00:43:19,350 --> 00:43:24,900 Pairs hanging on mirror planes that are 30 degrees apart and 688 00:43:24,900 --> 00:43:28,840 hanging disposed and pointing in different ways on the 689 00:43:28,840 --> 00:43:30,360 adjacent mirror planes. 690 00:43:30,360 --> 00:43:33,910 So this is one that we would call in international notation 691 00:43:33,910 --> 00:43:39,215 6mmm and in Schoenflies notation C6v. 692 00:43:44,070 --> 00:43:48,270 So we're making extraordinary progress here. 693 00:43:48,270 --> 00:43:53,370 The one that I did for you initially was C2v, and that's 694 00:43:53,370 --> 00:43:57,300 2mn in international notation. 695 00:43:57,300 --> 00:44:02,030 The only one that I left out to this point is a threefold 696 00:44:02,030 --> 00:44:04,070 axis compared with a vertical mirror planes. 697 00:44:08,440 --> 00:44:20,370 And let's start with the operation A 2 pi/3, combined 698 00:44:20,370 --> 00:44:24,050 with that a first reflection operation sigma 1 that passes 699 00:44:24,050 --> 00:44:25,300 through it. 700 00:44:27,620 --> 00:44:32,380 And for reference, I'll drawn in some lines that are 701 00:44:32,380 --> 00:44:36,690 separated by intervals of 60 degrees. 702 00:44:36,690 --> 00:44:42,610 So this reflection from 1 to 2 followed by a rotation 120 703 00:44:42,610 --> 00:44:46,740 degrees should give me this one as number 3. 704 00:44:46,740 --> 00:44:48,280 The first is right handed. 705 00:44:48,280 --> 00:44:54,360 The second is left handed, and the third stays left handed. 706 00:44:54,360 --> 00:44:59,950 And lo and behold, the way I get from 1 to 3 directly is by 707 00:44:59,950 --> 00:45:07,450 a reflection sigma 2 across a mirror line that is-- 708 00:45:07,450 --> 00:45:08,310 I'm sorry. 709 00:45:08,310 --> 00:45:13,190 This is number 1 down here-- across a mirror line that is 710 00:45:13,190 --> 00:45:16,504 30 degrees away from the first. 711 00:45:16,504 --> 00:45:18,970 If I would complete the pattern, reflect this one 712 00:45:18,970 --> 00:45:24,840 across to here, take this pair and reflect it or rotate it up 713 00:45:24,840 --> 00:45:29,440 here, and I have six objects. 714 00:45:29,440 --> 00:45:31,610 So this is a group of rank 6. 715 00:45:31,610 --> 00:45:34,980 And it has characteristics that are quite analogous to 716 00:45:34,980 --> 00:45:39,220 those we did for the other rotation axes in all respect 717 00:45:39,220 --> 00:45:41,520 except one. 718 00:45:41,520 --> 00:45:45,915 The international symbol for this combination is C3v. 719 00:45:48,560 --> 00:45:52,090 We've got a threefold axis. 720 00:45:52,090 --> 00:45:55,060 We've got the mirror plane that we added. 721 00:45:55,060 --> 00:45:59,510 And then we've got a mirror plane that is 30 degrees away 722 00:45:59,510 --> 00:46:00,155 from the first-- 723 00:46:00,155 --> 00:46:01,310 I'm sorry-- 724 00:46:01,310 --> 00:46:03,810 60 degrees away from the first, 725 00:46:03,810 --> 00:46:05,370 one half of 120 degrees. 726 00:46:10,570 --> 00:46:15,700 I claim that this is not the proper symbol because the 727 00:46:15,700 --> 00:46:17,710 mirror planes that are listed should be the 728 00:46:17,710 --> 00:46:22,110 symmetry-independent, distinct sort of mirror planes. 729 00:46:22,110 --> 00:46:25,890 And is that the case with this one? 730 00:46:29,110 --> 00:46:29,960 The answer is no. 731 00:46:29,960 --> 00:46:31,230 It isn't. 732 00:46:31,230 --> 00:46:32,550 Here is one mirror plane. 733 00:46:32,550 --> 00:46:34,750 It's got a pair of motifs hanging on it. 734 00:46:34,750 --> 00:46:38,360 This mirror plane here is a mirror plane that has a pair 735 00:46:38,360 --> 00:46:40,270 of motifs hanging it. 736 00:46:40,270 --> 00:46:42,530 Same is true of this mirror plane. 737 00:46:42,530 --> 00:46:46,880 So all six of these mirror planes are doing the same 738 00:46:46,880 --> 00:46:48,130 thing in the pattern. 739 00:46:48,130 --> 00:46:51,020 Each one has a pair of a motifs on either side of it in 740 00:46:51,020 --> 00:46:52,890 the same fashion. 741 00:46:52,890 --> 00:46:56,850 And I can get one mirror plane and the motifs hanging on it 742 00:46:56,850 --> 00:47:00,740 by a rotation of 120 degrees. 743 00:47:00,740 --> 00:47:03,780 So there's only one kind of mirror planes, so we don't 744 00:47:03,780 --> 00:47:05,570 need that m. 745 00:47:05,570 --> 00:47:10,076 So all of the other rotational symmetries, m, a onefold axis 746 00:47:10,076 --> 00:47:14,090 with a mirror plane; 2mm, 4mm, 6mm have two 747 00:47:14,090 --> 00:47:16,970 kinds of mirror planes. 748 00:47:16,970 --> 00:47:21,670 C3v has only one kind of mirror planes. 749 00:47:21,670 --> 00:47:23,830 And another way of showing that is if I look at a 750 00:47:23,830 --> 00:47:26,960 trigonal prism. 751 00:47:26,960 --> 00:47:29,800 It's got a mirror plane coming out of a corner and out of the 752 00:47:29,800 --> 00:47:31,410 opposite face. 753 00:47:31,410 --> 00:47:34,580 Mirror plane coming out of the corner and out 754 00:47:34,580 --> 00:47:35,920 the opposite face. 755 00:47:35,920 --> 00:47:38,400 Mirror plane coming out of a corner and out 756 00:47:38,400 --> 00:47:39,750 the opposite face. 757 00:47:39,750 --> 00:47:41,920 And that's exactly the rearrangement of symmetry 758 00:47:41,920 --> 00:47:43,030 elements here. 759 00:47:43,030 --> 00:47:46,620 So each mirror planes when drawn in relative to a 760 00:47:46,620 --> 00:47:50,560 trigonal prism, which is a body that has this symmetry, 761 00:47:50,560 --> 00:47:53,510 each mirror planes does exactly the same thing, as 762 00:47:53,510 --> 00:47:57,340 opposed to the square tile or square prism that has one kind 763 00:47:57,340 --> 00:48:00,100 of mirror plane that comes out of faces and one kind of 764 00:48:00,100 --> 00:48:01,850 mirror plane that comes out of corners. 765 00:48:01,850 --> 00:48:03,500 So there are two different types of mirror planes. 766 00:48:10,500 --> 00:48:18,740 Let's add them all up, summarize, and we've got the 767 00:48:18,740 --> 00:48:23,730 cast of characters that we can be use in deriving 768 00:48:23,730 --> 00:48:25,920 two-dimensional symmetries. 769 00:48:25,920 --> 00:48:30,650 We've got C1, which is a onefold axis; C2, twofold 770 00:48:30,650 --> 00:48:36,950 axis; C3, that's a threefold axis; C4, and that's a 771 00:48:36,950 --> 00:48:42,550 fourfold axis; C6, that's a sixfold axis. 772 00:48:42,550 --> 00:48:46,690 Then we've got the additions that involve adding a mirror 773 00:48:46,690 --> 00:48:48,640 plane to a rotational axis. 774 00:48:48,640 --> 00:48:55,890 So this is simply m and CS in Schoenflies notation. 775 00:48:55,890 --> 00:48:58,460 Then we added a mirror plane to a twofold axis. 776 00:48:58,460 --> 00:49:07,160 We've got C2 with a vertical mirror plane, 2mm; C3v; and 777 00:49:07,160 --> 00:49:18,180 not 3mm, but only 3m; C4v, and that's 4mm; and C6v, and 778 00:49:18,180 --> 00:49:20,990 that's 6mm. 779 00:49:20,990 --> 00:49:24,890 Add them all up, and there are 10 unique possibilities, no 780 00:49:24,890 --> 00:49:26,480 more, no less. 781 00:49:26,480 --> 00:49:26,650 Yes, sir? 782 00:49:26,650 --> 00:49:29,090 AUDIENCE: I'm missing a point. 783 00:49:29,090 --> 00:49:33,810 Why is there the mirror plane sigma 2 rather than rotating 784 00:49:33,810 --> 00:49:36,820 sigma 1 is not different? 785 00:49:36,820 --> 00:49:38,290 How do you? 786 00:49:38,290 --> 00:49:40,690 PROFESSOR: The same thing is going on, on this mirror plane 787 00:49:40,690 --> 00:49:43,730 as on this mirror planes, on this mirror plane. 788 00:49:43,730 --> 00:49:48,120 Or if you like to see it in terms of a geometric solid, 789 00:49:48,120 --> 00:49:52,420 the mirror planes here in a trigonal prism all of them do 790 00:49:52,420 --> 00:49:53,010 the same thing. 791 00:49:53,010 --> 00:49:57,360 They come out of one corner and out of the opposite edge. 792 00:49:57,360 --> 00:50:00,770 Each of these has one pair of objects hanging on it. 793 00:50:00,770 --> 00:50:04,800 And that is true of all three of them. 794 00:50:04,800 --> 00:50:11,000 And if you like, this end of the mirror plane here is 795 00:50:11,000 --> 00:50:15,380 nothing more than one that's related to the first one by 796 00:50:15,380 --> 00:50:18,550 one of the rotations of the threefold axis extended back 797 00:50:18,550 --> 00:50:20,340 in the opposite direction. 798 00:50:20,340 --> 00:50:22,290 So the mirror plane doesn't just work up here. 799 00:50:22,290 --> 00:50:23,480 It works down here as well. 800 00:50:23,480 --> 00:50:25,790 It works all through the space. 801 00:50:25,790 --> 00:50:28,630 So any way you want to look at it and whatever terms work for 802 00:50:28,630 --> 00:50:32,900 you, each of these mirror planes is the same, but they 803 00:50:32,900 --> 00:50:35,860 have a different type of end to them. 804 00:50:35,860 --> 00:50:37,720 There's something different hanging at one end than 805 00:50:37,720 --> 00:50:41,230 at the other end. 806 00:50:41,230 --> 00:50:43,510 Think of it in terms of actual physical object. 807 00:50:43,510 --> 00:50:45,990 AUDIENCE: [? What about the ?] case of the fourfold 808 00:50:45,990 --> 00:50:46,982 [? there? ?] 809 00:50:46,982 --> 00:50:51,190 PROFESSOR: In the fourfold, one comes out of the face. 810 00:50:51,190 --> 00:50:53,210 One comes out at the edge. 811 00:50:53,210 --> 00:50:58,360 The motifs are oriented and spaced at a different distance 812 00:50:58,360 --> 00:51:00,005 from each of those neighboring mirror planes. 813 00:51:09,650 --> 00:51:10,950 So these two are different. 814 00:51:10,950 --> 00:51:12,970 The fourfold axis never rotates this 815 00:51:12,970 --> 00:51:15,410 one into this one. 816 00:51:15,410 --> 00:51:19,790 These two are not different because the threefold axis 817 00:51:19,790 --> 00:51:25,990 rotates, if you do it twice, this end into this end of what 818 00:51:25,990 --> 00:51:29,300 is one in the same mirror plane as what's down on here. 819 00:51:29,300 --> 00:51:32,190 So what's confusing you I think is that there's a 820 00:51:32,190 --> 00:51:35,170 polarity to the mirror planes. 821 00:51:35,170 --> 00:51:37,930 Both ends of the mirror planes do not have the same 822 00:51:37,930 --> 00:51:41,990 disposition of motifs on them, but there's nothing that says 823 00:51:41,990 --> 00:51:43,650 that this has to be the case. 824 00:51:43,650 --> 00:51:46,630 The motifs could just be at one end. 825 00:51:46,630 --> 00:51:50,360 And that, if you think about it a little bit if it help, 826 00:51:50,360 --> 00:51:51,660 figure out-- 827 00:51:51,660 --> 00:51:54,180 not on company time though, but on your own time-- what a 828 00:51:54,180 --> 00:51:56,390 fivefold axis does. 829 00:51:56,390 --> 00:51:59,350 And a fivefold axis is a non crystallographic symmetry. 830 00:51:59,350 --> 00:52:09,450 But it turns out that C5v, which is a 5 with an m, has 831 00:52:09,450 --> 00:52:13,870 only one kind of mirror plane in it as well and for exactly 832 00:52:13,870 --> 00:52:14,610 the same reason. 833 00:52:14,610 --> 00:52:17,670 A regular figure that has the symmetry is a pentagon. 834 00:52:17,670 --> 00:52:20,300 A mirror plane comes out a corner and out of 835 00:52:20,300 --> 00:52:21,860 the opposite face. 836 00:52:21,860 --> 00:52:25,210 And that's true for all of the mirror planes that are in 837 00:52:25,210 --> 00:52:30,000 there and separated by one half of 2 pi/5. 838 00:52:30,000 --> 00:52:31,920 AUDIENCE: That's just whether or not the [INAUDIBLE] axis is 839 00:52:31,920 --> 00:52:32,400 [INAUDIBLE]? 840 00:52:32,400 --> 00:52:32,980 PROFESSOR: Yeah. 841 00:52:32,980 --> 00:52:33,965 That's what I said. 842 00:52:33,965 --> 00:52:36,380 Yeah. 843 00:52:36,380 --> 00:52:40,910 So before I go through all of the rotation axes with 844 00:52:40,910 --> 00:52:43,577 vertical mirror planes, the international notation would 845 00:52:43,577 --> 00:52:53,450 be 2mm, 3m, 4mm, 5m, and 6mm, 7m. 846 00:52:53,450 --> 00:52:55,850 For the odd symmetries, there's only one kind of 847 00:52:55,850 --> 00:52:59,760 mirror plane in a pattern, in the tile, or 848 00:52:59,760 --> 00:53:01,890 whatever you want to have. 849 00:53:01,890 --> 00:53:06,680 So there are 10 distinct possibilities, and these are 850 00:53:06,680 --> 00:53:09,440 called the Point Groups. 851 00:53:15,410 --> 00:53:16,200 Why? 852 00:53:16,200 --> 00:53:21,050 Because they are clusters of symmetry elements about at 853 00:53:21,050 --> 00:53:23,900 least one point that's fixed and embedded 854 00:53:23,900 --> 00:53:25,150 immovably in space. 855 00:53:28,890 --> 00:53:32,150 They're called groups because, as we've seen for one simple 856 00:53:32,150 --> 00:53:37,620 example, the collection of operations follows the 857 00:53:37,620 --> 00:53:40,810 postulates for the set of elements which we have defined 858 00:53:40,810 --> 00:53:42,060 as a group. 859 00:53:44,710 --> 00:53:47,880 More specifically, we could call these the 10 860 00:53:47,880 --> 00:53:50,750 two-dimensional crystallographic graphic Point 861 00:53:50,750 --> 00:54:01,840 Groups, which is more of a mouthful. 862 00:54:01,840 --> 00:54:04,360 But it emphasizes the fact that there are lots of 863 00:54:04,360 --> 00:54:07,790 two-dimensional Point Groups, but the two-dimensional 864 00:54:07,790 --> 00:54:11,470 crystallographic Point Groups are those that involve 865 00:54:11,470 --> 00:54:14,930 rotational symmetry that are compatible with a lattice. 866 00:54:14,930 --> 00:54:18,210 Hence, they are crystallographic. 867 00:54:18,210 --> 00:54:21,950 But the number of Point Groups is actually infinite if you 868 00:54:21,950 --> 00:54:27,187 include the ones that are not compatible with translation. 869 00:54:27,187 --> 00:54:28,660 All right. 870 00:54:28,660 --> 00:54:31,070 I think that's probably a good place to quit. 871 00:54:31,070 --> 00:54:31,770 And guess what? 872 00:54:31,770 --> 00:54:34,470 My internal clock has told me that this is the 873 00:54:34,470 --> 00:54:36,440 time for our break. 874 00:54:36,440 --> 00:54:42,140 This is one crossroads in our development. 875 00:54:42,140 --> 00:54:47,180 And what we'll do next for the faint hearted who may not want 876 00:54:47,180 --> 00:54:49,510 to have more of this stuff for one day than what we've just 877 00:54:49,510 --> 00:54:54,020 done, we're now going to do the final penultimate 878 00:54:54,020 --> 00:54:59,790 combination, and say we have also shown that there are 5 879 00:54:59,790 --> 00:55:01,040 two-dimensional lattices. 880 00:55:08,710 --> 00:55:11,370 And the final step will be to say if we have a pattern that 881 00:55:11,370 --> 00:55:15,170 has symmetry and is based on translation, we can obtain 882 00:55:15,170 --> 00:55:19,530 these by taking each of the 10 crystallographic Point Groups 883 00:55:19,530 --> 00:55:27,270 in turn and dropping them into each of the five lattices that 884 00:55:27,270 --> 00:55:28,560 can accommodate them. 885 00:55:28,560 --> 00:55:32,630 We would not try, for example, to take a Point Group like 4mm 886 00:55:32,630 --> 00:55:34,880 and try to drop it into the hexagonal lattice. 887 00:55:34,880 --> 00:55:36,230 It's not going to fit. 888 00:55:36,230 --> 00:55:41,660 But there will be two, maybe three ways in which we can add 889 00:55:41,660 --> 00:55:44,250 a given Point Group to one of these lattice types. 890 00:55:44,250 --> 00:55:47,240 And what we've done when we finished is we have 891 00:55:47,240 --> 00:55:52,690 exhaustively derived the symmetries' translational 892 00:55:52,690 --> 00:55:57,010 lattices and symmetry operations of reflection and 893 00:55:57,010 --> 00:56:01,960 rotation that are possible for two-dimensional crystal. 894 00:56:01,960 --> 00:56:05,210 You might say, well two-dimensional crystal, I've 895 00:56:05,210 --> 00:56:08,310 heard that people who do thin film work to make monolayers 896 00:56:08,310 --> 00:56:09,950 and really make a two-dimensional crystal. 897 00:56:09,950 --> 00:56:12,710 That's not something you could say only a few years ago. 898 00:56:12,710 --> 00:56:16,350 But why do we worry about two-dimensional symmetries if 899 00:56:16,350 --> 00:56:18,840 we're not to be wallpaper designers? 900 00:56:18,840 --> 00:56:20,910 Well, actually, I'll give you one example. 901 00:56:20,910 --> 00:56:23,930 One of the difficulties you had in conveying the nature of 902 00:56:23,930 --> 00:56:26,370 a crystal structure is taking something that's fairly 903 00:56:26,370 --> 00:56:29,130 complicated in three dimensions and getting it onto 904 00:56:29,130 --> 00:56:30,790 a two-dimensional sheet of paper. 905 00:56:30,790 --> 00:56:35,120 And what you invariably do is you project the contents of 906 00:56:35,120 --> 00:56:38,170 the unit cell down along one of the cell edges. 907 00:56:38,170 --> 00:56:40,720 And what you have then is a two-dimensional crystal 908 00:56:40,720 --> 00:56:43,200 structure with a lattice and symmetry. 909 00:56:43,200 --> 00:56:46,860 So you'll see two-dimensional representations of 910 00:56:46,860 --> 00:56:49,540 translationally periodic arrangements of atoms quite 911 00:56:49,540 --> 00:56:52,280 frequently when you look at projections of actual crystal 912 00:56:52,280 --> 00:56:56,150 structures, which is the only reasonable way to convey the 913 00:56:56,150 --> 00:56:59,300 information of any structure once you get on beyond the 914 00:56:59,300 --> 00:57:02,910 baby stuff of sodium chlorides, zinc, sulfide and 915 00:57:02,910 --> 00:57:04,160 body-centered iron. 916 00:57:06,670 --> 00:57:08,060 OK. 917 00:57:08,060 --> 00:57:10,720 I'm going to pause, suck in air. 918 00:57:10,720 --> 00:57:16,130 I am happy to turn you loose for 10 minutes and will meet 919 00:57:16,130 --> 00:57:17,790 here again, let's say, at 10 after 3:00.