1 00:00:05,760 --> 00:00:08,910 PROFESSOR: Another strange observation about mirrors that 2 00:00:08,910 --> 00:00:10,310 I've never really understood-- 3 00:00:12,880 --> 00:00:16,520 the mirror plane is reflecting me left to right, so it looks 4 00:00:16,520 --> 00:00:20,680 as though a mirror has a grain to it. 5 00:00:20,680 --> 00:00:25,850 It knows what line to reflect me back and forth. 6 00:00:25,850 --> 00:00:31,420 But if I kept the mirror in fixed orientation and I lay 7 00:00:31,420 --> 00:00:34,900 down, the thing should reflect me side to 8 00:00:34,900 --> 00:00:37,400 side, but it doesn't. 9 00:00:37,400 --> 00:00:42,380 It still reflects me from top to bottom. 10 00:00:42,380 --> 00:00:44,080 So how can that be? 11 00:00:44,080 --> 00:00:47,050 Why does a mirror plane, if I hold it in one orientation, 12 00:00:47,050 --> 00:00:51,290 appear to have a direction across which I'm reflected but 13 00:00:51,290 --> 00:00:52,920 it doesn't follow me if I move? 14 00:00:56,890 --> 00:00:58,140 Know what I mean? 15 00:00:59,700 --> 00:01:01,862 You have any explanation of that? 16 00:01:01,862 --> 00:01:02,344 Hmm? 17 00:01:02,344 --> 00:01:04,272 AUDIENCE: Rotate your eyes, too. 18 00:01:04,272 --> 00:01:07,990 PROFESSOR: Rotate my eyes. 19 00:01:07,990 --> 00:01:09,280 I can roll them around. 20 00:01:09,280 --> 00:01:10,775 I can't rotate in any other fashion. 21 00:01:13,720 --> 00:01:14,340 That's strange. 22 00:01:14,340 --> 00:01:17,760 I mean, you look at yourself every morning-- 23 00:01:17,760 --> 00:01:20,200 several times, perhaps, and you're reflected 24 00:01:20,200 --> 00:01:21,820 always left to right. 25 00:01:21,820 --> 00:01:24,490 And if you turn the mirror, it doesn't 26 00:01:24,490 --> 00:01:27,080 reflect you top to bottom. 27 00:01:27,080 --> 00:01:29,740 Or conversely, you can leave the mirror alone and you can-- 28 00:01:29,740 --> 00:01:34,635 why does a mirror plane just reflect you left to right? 29 00:01:34,635 --> 00:01:37,450 Ah. 30 00:01:37,450 --> 00:01:41,390 I'll let you stew about that one for a while. 31 00:01:41,390 --> 00:01:47,720 And I can tell you that I know of three papers in scientific 32 00:01:47,720 --> 00:01:51,290 journals that tried to explain this. 33 00:01:51,290 --> 00:01:54,125 I can give you literature citations, but you think about 34 00:01:54,125 --> 00:01:58,080 it until our next meeting. 35 00:02:02,020 --> 00:02:04,310 Why does it know which way I'm oriented? 36 00:02:04,310 --> 00:02:06,320 Ah, I can't think about it. 37 00:02:06,320 --> 00:02:11,090 Alright, back to more straightforward questions. 38 00:02:11,090 --> 00:02:16,360 We are now about to embark on a grand process of synthesis 39 00:02:16,360 --> 00:02:21,240 which will take us the better part of half a semester. 40 00:02:21,240 --> 00:02:25,690 And we've identified our four basic operations-- 41 00:02:25,690 --> 00:02:28,000 four basic one step operations-- 42 00:02:28,000 --> 00:02:41,760 namely translation, reflection, rotation, and for 43 00:02:41,760 --> 00:02:44,750 the time being I'm going to look just at two-dimensional 44 00:02:44,750 --> 00:02:48,240 symmetries, so I'll leave inversion out of the picture. 45 00:02:48,240 --> 00:02:51,990 It's only defined in three dimensions, and the logic 46 00:02:51,990 --> 00:02:55,970 which I will follow will be to first build up two-dimensional 47 00:02:55,970 --> 00:02:58,840 symmetries and then we'll turn them into three-dimensional 48 00:02:58,840 --> 00:03:02,900 symmetries by picking another translation that's not 49 00:03:02,900 --> 00:03:05,080 coplanar with the first two. 50 00:03:05,080 --> 00:03:10,110 So we're going to look for the time being at just 51 00:03:10,110 --> 00:03:12,910 two-dimensional symmetries. 52 00:03:12,910 --> 00:03:16,810 The nice thing about doing this is that the number of 53 00:03:16,810 --> 00:03:20,680 two-dimensional symmetries is relatively small, so we can 54 00:03:20,680 --> 00:03:23,470 derive them rigorously and exhaustively. 55 00:03:23,470 --> 00:03:26,720 To do so in three dimensions is a much more time consuming 56 00:03:26,720 --> 00:03:28,400 and elaborate exercise. 57 00:03:28,400 --> 00:03:29,880 It's no different in principle, 58 00:03:29,880 --> 00:03:31,380 there's just more work. 59 00:03:31,380 --> 00:03:33,710 So if we do two-dimensional symmetries first, 60 00:03:33,710 --> 00:03:34,690 it's an easy case. 61 00:03:34,690 --> 00:03:37,370 We can do it rigorously and completely, and then what 62 00:03:37,370 --> 00:03:41,140 we'll do is just look at a few examples in three dimensions 63 00:03:41,140 --> 00:03:47,050 and look at how the results are designated and tabulated. 64 00:03:47,050 --> 00:03:50,410 So the first combination I will choose to make is one 65 00:03:50,410 --> 00:03:53,600 that I set up last time and should not have started when 66 00:03:53,600 --> 00:03:55,100 there was no time to finish it. 67 00:03:55,100 --> 00:04:00,370 So let me take an initially pristine space and say that to 68 00:04:00,370 --> 00:04:03,640 this I'm going to the operation of translation. 69 00:04:03,640 --> 00:04:07,120 That immediately implies a string of translations and a 70 00:04:07,120 --> 00:04:09,740 string of lattice points, but I'm just going to focus my 71 00:04:09,740 --> 00:04:11,370 attention on the first one. 72 00:04:14,660 --> 00:04:21,220 Then what I said I'll do is add to the space a rotation 73 00:04:21,220 --> 00:04:23,100 operation A alpha. 74 00:04:23,100 --> 00:04:26,010 I'm going to for convenience put it right at the lattice 75 00:04:26,010 --> 00:04:29,420 point, but as there is no unique origin to the 76 00:04:29,420 --> 00:04:31,350 translation, I can start and stop the 77 00:04:31,350 --> 00:04:33,520 translation anywhere I like. 78 00:04:33,520 --> 00:04:36,520 So here's my translation. 79 00:04:36,520 --> 00:04:40,030 The rotation operation is going to take that translation 80 00:04:40,030 --> 00:04:43,720 and repeat it at angular intervals alpha so that I get 81 00:04:43,720 --> 00:04:48,540 a radiating porcupine-like sheaf of translations all 82 00:04:48,540 --> 00:04:51,280 coming out of a common point. 83 00:04:51,280 --> 00:04:55,790 Alpha, we observed, has to be a submultiple of 2 pi, so I 84 00:04:55,790 --> 00:04:59,390 will have a cluster of translations separated by the 85 00:04:59,390 --> 00:05:01,220 equal interval alpha-- 86 00:05:01,220 --> 00:05:02,310 angular interval alpha-- 87 00:05:02,310 --> 00:05:06,560 and I'm going to choose to focus my attention on 88 00:05:06,560 --> 00:05:08,340 just the first one. 89 00:05:08,340 --> 00:05:11,470 So this, since it's repeated by rotation, has the same 90 00:05:11,470 --> 00:05:13,180 length T. 91 00:05:13,180 --> 00:05:15,380 And then I'm going to look at the other end of the 92 00:05:15,380 --> 00:05:19,840 translation and say that similarly I must have a set of 93 00:05:19,840 --> 00:05:25,200 equally spaced angular-wise translations all separated by 94 00:05:25,200 --> 00:05:28,290 alpha, and I'm going to choose to focus my attention 95 00:05:28,290 --> 00:05:29,830 just on this one. 96 00:05:29,830 --> 00:05:35,670 So this angle is alpha, and this angle here is alpha. 97 00:05:35,670 --> 00:05:38,450 And now there's big trouble in River City. 98 00:05:38,450 --> 00:05:40,800 This is a translation, here's a lattice point, here's a 99 00:05:40,800 --> 00:05:41,460 lattice point. 100 00:05:41,460 --> 00:05:42,710 This is a translation-- 101 00:05:42,710 --> 00:05:44,550 a lattice point sits up here. 102 00:05:44,550 --> 00:05:46,210 This is a translation-- 103 00:05:46,210 --> 00:05:48,800 a lattice points sits up here. 104 00:05:48,800 --> 00:05:52,370 Now I've got two lattice points eyeball to eyeball, and 105 00:05:52,370 --> 00:05:56,100 they jolly well better be separated by either the 106 00:05:56,100 --> 00:06:02,780 interval T or some multiple P times T, 107 00:06:02,780 --> 00:06:05,726 where P is some integer. 108 00:06:05,726 --> 00:06:08,910 It would be quite all right if there were two translation 109 00:06:08,910 --> 00:06:14,220 separation or five translation separation, but it must be 110 00:06:14,220 --> 00:06:18,020 some multiple of translation or I have violated my initial 111 00:06:18,020 --> 00:06:21,790 premise that everything in this space is periodic at a 112 00:06:21,790 --> 00:06:25,460 translational interval T. 113 00:06:25,460 --> 00:06:29,300 So let me take this geometrical constraint and 114 00:06:29,300 --> 00:06:33,770 very quickly convert it into analytical form so that we can 115 00:06:33,770 --> 00:06:37,300 proceed to systematically find out what the 116 00:06:37,300 --> 00:06:39,300 possibilities are. 117 00:06:39,300 --> 00:06:41,780 Let me drop a perpendicular down to the original 118 00:06:41,780 --> 00:06:44,160 translation. 119 00:06:44,160 --> 00:06:50,220 So this distance in here is PT, this distance will be T 120 00:06:50,220 --> 00:06:57,480 times the cosine of alpha, and this distance in here will be 121 00:06:57,480 --> 00:07:01,700 T times the cosine of alpha. 122 00:07:01,700 --> 00:07:05,580 So in analytic form, then, I can say that my original 123 00:07:05,580 --> 00:07:12,020 translation T is equal to T cosine of alpha 124 00:07:12,020 --> 00:07:13,590 plus T cosine of alpha. 125 00:07:13,590 --> 00:07:17,270 That's 2T cosine of alpha plus PT. 126 00:07:20,350 --> 00:07:26,130 And the first thing we can see is that the magnitude of T 127 00:07:26,130 --> 00:07:29,090 drops out because this construction and the 128 00:07:29,090 --> 00:07:32,580 constraint it embodies in no way depends on the magnitude 129 00:07:32,580 --> 00:07:33,790 of the translation. 130 00:07:33,790 --> 00:07:38,870 So this says, then, that 1 is equal to 2 cosine of alpha 131 00:07:38,870 --> 00:07:54,220 plus the integer P. And if I solve for the value of cosine 132 00:07:54,220 --> 00:07:59,900 of alpha, cosine of alpha will be 1 minus an integer P 133 00:07:59,900 --> 00:08:03,300 divided by 2. 134 00:08:03,300 --> 00:08:07,460 So there is the constraint that must be followed if my 135 00:08:07,460 --> 00:08:10,940 construction is to be self-consistent. 136 00:08:10,940 --> 00:08:14,640 So we've got this now in a plug and chug situation. 137 00:08:14,640 --> 00:08:18,850 So what I'm going to put down is the value of P, taking all 138 00:08:18,850 --> 00:08:21,310 possible integers for which a value of 139 00:08:21,310 --> 00:08:23,910 cosine of alpha exists. 140 00:08:23,910 --> 00:08:27,380 I'm then going to evaluate cosine of alpha, which is 1 141 00:08:27,380 --> 00:08:34,210 minus P over 2, and then I'm going to identify the n-fold 142 00:08:34,210 --> 00:08:37,640 axis that corresponds to that particular value of alpha. 143 00:08:42,200 --> 00:08:46,640 Let's put in P equals 3, and this is as far 144 00:08:46,640 --> 00:08:48,170 as we got last time. 145 00:08:48,170 --> 00:08:52,380 Well, if we put 4, then cosine of alpha is minus 3/2-- 146 00:08:52,380 --> 00:08:53,970 it's not defined. 147 00:08:53,970 --> 00:09:00,650 If we drop P down to 3, then cosine of alpha is minus 1, 1 148 00:09:00,650 --> 00:09:03,020 minus 3 over 2. 149 00:09:03,020 --> 00:09:08,405 And the angle whose cosine is minus 1 is a-- 150 00:09:08,405 --> 00:09:10,370 let me put down the value of alpha and 151 00:09:10,370 --> 00:09:13,700 then the n-fold axis. 152 00:09:13,700 --> 00:09:19,080 The angle whose cosine is minus 1 is 180 degrees, and 153 00:09:19,080 --> 00:09:24,050 that would describe quite nicely the rotational throw of 154 00:09:24,050 --> 00:09:26,680 a twofold axis. 155 00:09:26,680 --> 00:09:34,130 If I let P drop down to the value 2, then I have minus 1/2 156 00:09:34,130 --> 00:09:36,130 of the cosine of alpha. 157 00:09:36,130 --> 00:09:42,530 The angle whose cosine is minus 1/2 is 120 degrees. 158 00:09:42,530 --> 00:09:43,150 And guess what? 159 00:09:43,150 --> 00:09:45,350 That's the angular throw of a threefold axis. 160 00:09:48,390 --> 00:09:53,720 Let P drop down to 1, and then cosine of alpha is 0. 161 00:09:53,720 --> 00:09:59,270 The angle whose cosine is 0 is 90 degrees, and that would be 162 00:09:59,270 --> 00:10:00,520 a fourfold axis. 163 00:10:03,230 --> 00:10:10,570 Looks like we're done, except P could be equal to 0. 164 00:10:10,570 --> 00:10:16,920 In that case, cosine of alpha is plus 1/2. 165 00:10:16,920 --> 00:10:25,740 The angle whose cosine is plus 1/2 is equal to 60 degrees, 166 00:10:25,740 --> 00:10:28,460 and that's a sixfold access. 167 00:10:28,460 --> 00:10:30,850 What about negative integers? 168 00:10:30,850 --> 00:10:33,290 Minus one, will that work? 169 00:10:33,290 --> 00:10:38,710 This says that cosine of alpha is 1, and the angle whose 170 00:10:38,710 --> 00:10:42,960 cosine is one is 0 degrees or 360 degrees. 171 00:10:42,960 --> 00:10:45,460 And that would be no rotational symmetry all. 172 00:10:45,460 --> 00:10:49,070 It's rather amusing that the trivial case of no symmetry at 173 00:10:49,070 --> 00:10:53,080 all also falls out of this construction. 174 00:10:53,080 --> 00:10:58,040 So this is a momentous result. 175 00:10:58,040 --> 00:11:03,010 We've shown that if you're going to have a pattern that 176 00:11:03,010 --> 00:11:06,900 has a repetition by translation in it, the number 177 00:11:06,900 --> 00:11:11,450 of rotational symmetries that can be added are either no 178 00:11:11,450 --> 00:11:16,850 symmetry at all, a twofold rotational symmetry, 179 00:11:16,850 --> 00:11:22,340 threefold, fourfold, or sixfold. 180 00:11:26,340 --> 00:11:27,580 In other words, the axis-- 181 00:11:27,580 --> 00:11:32,810 no symmetry at all, twofold, threefold, fourfold, or 182 00:11:32,810 --> 00:11:35,700 sixfold, nothing else. 183 00:11:39,540 --> 00:11:42,380 This tells you a lot about the shapes that you can see 184 00:11:42,380 --> 00:11:44,330 macroscopically on crystals. 185 00:11:44,330 --> 00:11:47,920 You could have a crystal that had the shape of the trigonal 186 00:11:47,920 --> 00:11:53,820 prism, and that would be perfectly fine. 187 00:11:53,820 --> 00:11:57,550 You could have a crystal that had the shape of an orthogonal 188 00:11:57,550 --> 00:12:02,040 brick that would have twofold axes coming out of the faces. 189 00:12:02,040 --> 00:12:04,610 You could have a crystal in the shape of a hexagonal 190 00:12:04,610 --> 00:12:08,880 present, or you could have a crystal in the shape of the 191 00:12:08,880 --> 00:12:10,450 square prism. 192 00:12:10,450 --> 00:12:15,780 But something thing like a crystal with a pentagonal 193 00:12:15,780 --> 00:12:20,720 cross section that would be a fivefold access-- 194 00:12:20,720 --> 00:12:24,900 that is strictly forbidden, because the external shape of 195 00:12:24,900 --> 00:12:28,860 the crystal has to reflect the internal symmetry among the 196 00:12:28,860 --> 00:12:30,110 arrangement of atoms. 197 00:12:33,740 --> 00:12:39,750 There are lots of things in nature that have 198 00:12:39,750 --> 00:12:42,900 crystallographic symmetry. 199 00:12:42,900 --> 00:12:56,955 There is a little cactus that looks like this with some 200 00:12:56,955 --> 00:13:00,470 little spines coming out like this on the top, and its 201 00:13:00,470 --> 00:13:05,650 proper name is astrophytum myriostigma, also known as the 202 00:13:05,650 --> 00:13:07,980 ornamented bishops cap. 203 00:13:07,980 --> 00:13:11,970 Beautiful example of fivefold symmetry, but the cells inside 204 00:13:11,970 --> 00:13:16,290 of that cactus can not have the same size and shape and be 205 00:13:16,290 --> 00:13:18,050 repeated by translation. 206 00:13:20,710 --> 00:13:27,010 There are flowers that have very common examples of 207 00:13:27,010 --> 00:13:31,490 fivefold and even sevenfold symmetry. 208 00:13:31,490 --> 00:13:33,850 There's one little purple flower that looks like this 209 00:13:33,850 --> 00:13:38,370 that comes out in the spring, and that's called periwinkle. 210 00:13:43,170 --> 00:13:46,760 Fivefold symmetry-- fine for a plant, but if you got down 211 00:13:46,760 --> 00:13:51,370 inside the stem of this flower, the cells cannot be 212 00:13:51,370 --> 00:13:54,234 repeated by translation. 213 00:13:54,234 --> 00:13:57,770 There are astronomically high symmetries. 214 00:13:57,770 --> 00:14:05,250 These big giant cacti in the Southwest, the saguaro, they 215 00:14:05,250 --> 00:14:09,970 have rotational symmetries that run from 18-fold, 216 00:14:09,970 --> 00:14:14,740 19-fold, 23-fold, so that if you picked up one of these 217 00:14:14,740 --> 00:14:19,030 guys very carefully, because they're covered with spines, 218 00:14:19,030 --> 00:14:21,400 and if you could lift it-- which would be very hard 219 00:14:21,400 --> 00:14:23,460 because they weigh a couple of tons-- 220 00:14:23,460 --> 00:14:27,680 take that guy and rotate them through 1/27 of a circle, and 221 00:14:27,680 --> 00:14:30,930 if he had 27-fold symmetry, you could plop them down and 222 00:14:30,930 --> 00:14:33,930 you couldn't tell that it had been moved. 223 00:14:33,930 --> 00:14:39,060 There is no crystallographer who can resist cacti. 224 00:14:39,060 --> 00:14:42,830 All sorts of symmetry, even symmetries that violate 225 00:14:42,830 --> 00:14:44,680 crystal graphic symmetries-- 226 00:14:44,680 --> 00:14:47,680 wonderful textures and colors. 227 00:14:47,680 --> 00:14:53,190 And they have another really remarkable property, which 228 00:14:53,190 --> 00:14:54,540 commends them as house plants. 229 00:14:54,540 --> 00:15:00,420 If they die, this sheath of spines stays intact until 230 00:15:00,420 --> 00:15:02,990 someday when you're watering it, you brush against it and 231 00:15:02,990 --> 00:15:05,230 you poke a hole right through the spines and 232 00:15:05,230 --> 00:15:07,160 there's nothing inside. 233 00:15:07,160 --> 00:15:11,500 So a house plant that dies and you can't tell for two years 234 00:15:11,500 --> 00:15:17,230 or so is a very good plant to have as a companion. 235 00:15:17,230 --> 00:15:21,740 So cacti have all sorts of strange symmetries. 236 00:15:21,740 --> 00:15:24,420 Fine, but you can say something about the internal 237 00:15:24,420 --> 00:15:26,876 structure of that flower or that cactus. 238 00:15:29,990 --> 00:15:36,390 But this little almost trivial proof has told us something 239 00:15:36,390 --> 00:15:41,140 else about crystals, because the presence of translation 240 00:15:41,140 --> 00:15:44,520 imposes a constraint on the rotational symmetry that can 241 00:15:44,520 --> 00:15:52,730 be present, and the rotational symmetry tells us something 242 00:15:52,730 --> 00:15:55,930 about the nature of the two-dimensional lattice which 243 00:15:55,930 --> 00:15:57,650 can accommodate that symmetry. 244 00:15:57,650 --> 00:16:01,330 So let's go through this list once more, and let's pay 245 00:16:01,330 --> 00:16:04,790 attention to the value of P. 246 00:16:04,790 --> 00:16:10,160 For a twofold rotational symmetry, what we would do 247 00:16:10,160 --> 00:16:13,880 would be to take this original translation, we put the 248 00:16:13,880 --> 00:16:18,300 twofold axis here, and that takes the translation and 249 00:16:18,300 --> 00:16:19,610 rotates it around. 250 00:16:19,610 --> 00:16:23,540 So here's a lattice point here, one here, one here, and 251 00:16:23,540 --> 00:16:27,110 the twofold axis here, takes the translation and we rotate 252 00:16:27,110 --> 00:16:29,080 in a counterclockwise sense. 253 00:16:29,080 --> 00:16:31,360 Here's another lattice point here. 254 00:16:31,360 --> 00:16:37,460 P was equal to three translations, and I'll be 255 00:16:37,460 --> 00:16:40,780 darned if that isn't exactly what we have. 256 00:16:40,780 --> 00:16:44,005 Three translations from this lattice point A-- 257 00:16:44,005 --> 00:16:45,960 let me put some labels up here. 258 00:16:45,960 --> 00:16:51,740 This was lattice point A, this was lattice point B, and this 259 00:16:51,740 --> 00:16:54,590 was our translation PT. 260 00:16:54,590 --> 00:16:58,170 Three translations, just as advertised. 261 00:16:58,170 --> 00:17:02,530 What constraint does this put on a lattice? 262 00:17:02,530 --> 00:17:06,180 None whatsoever, because all this says is that if you have 263 00:17:06,180 --> 00:17:09,560 a translation that translation must be repeated into an 264 00:17:09,560 --> 00:17:12,369 extended one-dimensional row. 265 00:17:12,369 --> 00:17:21,089 So you can put this an any 2D lattice whatsoever. 266 00:17:21,089 --> 00:17:26,060 In other words, the magnitudes of the two translations-- 267 00:17:26,060 --> 00:17:30,780 let's call them T1 and T2 are under no constraint to be 268 00:17:30,780 --> 00:17:32,480 related one to another-- 269 00:17:32,480 --> 00:17:35,200 and this angle between them, alpha, can be 270 00:17:35,200 --> 00:17:36,450 anything that it likes. 271 00:17:39,040 --> 00:17:43,280 You could have a twofold axis, but that requires simply a 272 00:17:43,280 --> 00:17:46,760 lattice row parallel to T1 and a lattice row parallel to T2. 273 00:17:49,420 --> 00:17:52,020 The next integer we hit was minus 1/2. 274 00:17:55,200 --> 00:18:04,500 That was cosine of minus 1/2, this was P equals 3, and that 275 00:18:04,500 --> 00:18:08,280 corresponded to something that could 276 00:18:08,280 --> 00:18:14,435 accept a threefold symmetry. 277 00:18:14,435 --> 00:18:17,410 So let me put a guide to the I in here and make an 278 00:18:17,410 --> 00:18:18,970 equilateral triangle. 279 00:18:18,970 --> 00:18:25,920 If we translate and rotate up by 120 degrees using a 280 00:18:25,920 --> 00:18:30,910 threefold axis and then rotate minus 120 degrees about the 281 00:18:30,910 --> 00:18:33,790 other lattice point, that puts another translation 282 00:18:33,790 --> 00:18:37,550 up here, and now-- 283 00:18:37,550 --> 00:18:48,720 I'm sorry, this was P equals 2 And as required, this is PT, 284 00:18:48,720 --> 00:18:50,295 and that's equal to two translations. 285 00:18:55,300 --> 00:18:59,630 This has put a constraint on the sort of lattice which can 286 00:18:59,630 --> 00:19:05,490 exist in this space because we have two translations, T1 and 287 00:19:05,490 --> 00:19:09,650 T2, which are equal in magnitude, identical in 288 00:19:09,650 --> 00:19:12,780 magnitude, because they're related by symmetry. 289 00:19:12,780 --> 00:19:17,760 And consequently, we have defined a space lattice, a 290 00:19:17,760 --> 00:19:20,670 two-dimensional space lattice. 291 00:19:20,670 --> 00:19:23,940 We'll call this T1 and call this T2. 292 00:19:23,940 --> 00:19:27,660 This lattice has the constraint that magnitude of 293 00:19:27,660 --> 00:19:32,060 T1 must be identical to the magnitude of T2, and the angle 294 00:19:32,060 --> 00:19:39,170 between T1 and T2 is again identically 120 degrees. 295 00:19:39,170 --> 00:19:45,910 Not 119.9, but exactly 120 degrees because there is a 296 00:19:45,910 --> 00:19:50,630 threefold axis in here that demands that that be so. 297 00:19:50,630 --> 00:19:53,690 So this is a very specialized kind of lattice, restricted to 298 00:19:53,690 --> 00:19:57,533 have two translations identical in magnitude. 299 00:19:57,533 --> 00:20:00,520 And if there's a threefold axis there, you should be able 300 00:20:00,520 --> 00:20:01,860 to find these two. 301 00:20:01,860 --> 00:20:04,680 And if there's a threefold axis there, then the angle 302 00:20:04,680 --> 00:20:08,145 between these two specialized translations is 120 degrees. 303 00:20:22,720 --> 00:20:29,410 Our next magic integer was 1, and that corresponded to a 304 00:20:29,410 --> 00:20:30,660 fourfold axis. 305 00:20:33,890 --> 00:20:38,430 And if we do what we claimed we did in that construction, 306 00:20:38,430 --> 00:20:40,600 we'd put a fold axis here. 307 00:20:40,600 --> 00:20:45,690 That takes T1 and rotates it exactly 90 degrees to a 308 00:20:45,690 --> 00:20:47,540 translation T2. 309 00:20:47,540 --> 00:20:50,500 Once again, two noncollinear translations, so we have 310 00:20:50,500 --> 00:20:53,665 defined a two-dimensional net. 311 00:20:53,665 --> 00:20:58,630 If we complete the cell, this is one translation. 312 00:20:58,630 --> 00:21:05,670 PT is equal to one translation in here, and this angle is 90 313 00:21:05,670 --> 00:21:09,170 degrees because it's produced by a fourfold axis. 314 00:21:09,170 --> 00:21:11,310 So we have a very special lattice. 315 00:21:11,310 --> 00:21:15,680 Again, the magnitudes of two translations are identical-- 316 00:21:15,680 --> 00:21:18,380 not approximately the same, they're identical-- 317 00:21:18,380 --> 00:21:23,555 and the angle between them is identically 90 degrees. 318 00:21:30,830 --> 00:21:33,540 Only a couple to go. 319 00:21:33,540 --> 00:21:40,480 P could be equal to 0, and that was the case for a 60 320 00:21:40,480 --> 00:21:44,630 degree rotation, a sixfold axis. 321 00:21:44,630 --> 00:21:47,360 So again, let's draw what came out of this 322 00:21:47,360 --> 00:21:48,740 particular special case. 323 00:21:48,740 --> 00:21:53,810 Here's T1, here is T2. 324 00:21:53,810 --> 00:21:57,120 This angle is 60 degrees exactly because there's a 325 00:21:57,120 --> 00:22:01,660 six-fold rotation axis here, a sixfold rotation axis here, 326 00:22:01,660 --> 00:22:05,220 and the rotation of 60 in the opposite sense gives us 327 00:22:05,220 --> 00:22:06,960 another translation here. 328 00:22:06,960 --> 00:22:14,210 These two lattice points coincide and there is PT equal 329 00:22:14,210 --> 00:22:16,905 to 0T, and these two points coincide. 330 00:22:21,290 --> 00:22:26,490 Now if I complete a standard unit cell with T1 T2 as I've 331 00:22:26,490 --> 00:22:29,710 done in other cases-- 332 00:22:29,710 --> 00:22:36,200 this was T1, this was T2, this was a translation which I'm 333 00:22:36,200 --> 00:22:38,710 not going to use. 334 00:22:38,710 --> 00:22:42,320 So this is the shape of the lattice and these now are 335 00:22:42,320 --> 00:22:45,890 lattice points with a sixfold axis on them. 336 00:22:45,890 --> 00:22:50,600 The dimensional specialization is exactly the same as I found 337 00:22:50,600 --> 00:22:55,240 for a threefold axis, T1 identical to T2. 338 00:22:55,240 --> 00:23:01,130 If I pick this cell, T1 to T2 can be described as an angle 339 00:23:01,130 --> 00:23:02,405 of 120 degrees. 340 00:23:11,920 --> 00:23:16,910 Exactly the same lattice that we found for a threefold axis. 341 00:23:19,660 --> 00:23:22,690 So with this simple minded little construction we've 342 00:23:22,690 --> 00:23:25,410 found two profound things-- 343 00:23:25,410 --> 00:23:29,770 that there are five kinds of rotation axes, including the 344 00:23:29,770 --> 00:23:33,050 onefold no rotational symmetry at all. 345 00:23:33,050 --> 00:23:40,540 And it turns out that there are one, a general lattice, a 346 00:23:40,540 --> 00:23:43,420 hexagonal lattice, and the square lattice. 347 00:23:43,420 --> 00:23:46,380 There are three kinds of two-dimensional lattices that 348 00:23:46,380 --> 00:23:50,210 are required by these symmetry elements. 349 00:23:50,210 --> 00:23:58,440 So these guys require that there be three lattice of 350 00:23:58,440 --> 00:24:04,170 different specializations that are able to accommodate them. 351 00:24:04,170 --> 00:24:13,330 So let me call this a parallelogram net, and that 352 00:24:13,330 --> 00:24:17,910 has T1 not equal to T2, the angle 353 00:24:17,910 --> 00:24:20,620 between T1 and T2 general. 354 00:24:25,510 --> 00:24:29,380 And this is a lattice that can accommodate either no symmetry 355 00:24:29,380 --> 00:24:34,890 at all or a twofold rotation access. 356 00:24:34,890 --> 00:24:43,980 Then there was a net that I'll call the hexagonal net, and 357 00:24:43,980 --> 00:24:50,140 this had T1 identical to T2 in magnitude, and it had the 358 00:24:50,140 --> 00:24:54,570 angle between them, the angle between T1 and T2 as 359 00:24:54,570 --> 00:24:59,080 identically 120 degrees. 360 00:24:59,080 --> 00:25:02,330 And this could accommodate either a threefold or a 361 00:25:02,330 --> 00:25:05,390 sixfold axis. 362 00:25:05,390 --> 00:25:07,920 And then finally, the general net, which I call a 363 00:25:07,920 --> 00:25:18,470 parallelogram net, and that has magnitudes of the two 364 00:25:18,470 --> 00:25:22,430 translations not equal to one another. 365 00:25:22,430 --> 00:25:25,790 They can have any values they like, and the angle between T1 366 00:25:25,790 --> 00:25:30,590 and T2 is completely general. 367 00:25:30,590 --> 00:25:33,830 And that's exactly what I had up here for the-- oh, we did 368 00:25:33,830 --> 00:25:34,350 that once already. 369 00:25:34,350 --> 00:25:37,170 The one that I'm missing, the third one, is the square net. 370 00:25:43,910 --> 00:25:46,450 Square net has T1 identical to T2. 371 00:25:49,160 --> 00:25:55,670 In magnitude, the angle between them is exactly 90 372 00:25:55,670 --> 00:26:02,410 degrees, and that is required by a fourfold axis. 373 00:26:14,340 --> 00:26:16,715 Let me pause here to see if there are any questions. 374 00:26:19,600 --> 00:26:21,488 Yes, sir. 375 00:26:21,488 --> 00:26:24,890 AUDIENCE: When you write on the last one with the sixfold, 376 00:26:24,890 --> 00:26:26,348 it's only 120. 377 00:26:26,348 --> 00:26:28,778 You could have also written 60, correct? 378 00:26:28,778 --> 00:26:29,750 PROFESSOR: Yes, I could have. 379 00:26:29,750 --> 00:26:35,880 And this lattice is actually the same as what I found for a 380 00:26:35,880 --> 00:26:37,310 threefold axis. 381 00:26:37,310 --> 00:26:42,480 I could pick either this or this as the cell, but the two 382 00:26:42,480 --> 00:26:45,310 translations in those two cells are equal. 383 00:26:45,310 --> 00:26:50,070 And again at various stages along the way, we'll need a 384 00:26:50,070 --> 00:26:51,880 convention. 385 00:26:51,880 --> 00:26:56,700 And if I have a net that looks like this, a parallelogram, 386 00:26:56,700 --> 00:26:59,430 whether a specialized parallelogram or not, I have a 387 00:26:59,430 --> 00:27:02,520 choice of two angles that I could use. 388 00:27:02,520 --> 00:27:03,770 We call that alpha. 389 00:27:03,770 --> 00:27:07,930 This is going to be 180 degrees minus alpha. 390 00:27:07,930 --> 00:27:10,170 Which do I pick? 391 00:27:10,170 --> 00:27:21,040 You need a rule, and the rule is that for labeling the cell, 392 00:27:21,040 --> 00:27:25,920 pick alpha so that it's greater 393 00:27:25,920 --> 00:27:29,220 or equal to 90 degrees. 394 00:27:29,220 --> 00:27:32,230 That's pure convention, but you want to have a rule just 395 00:27:32,230 --> 00:27:35,810 like a language so people use the same words to describe the 396 00:27:35,810 --> 00:27:36,330 same thing. 397 00:27:36,330 --> 00:27:39,350 Here we want to use the same geometry to describe one and 398 00:27:39,350 --> 00:27:40,500 the same thing. 399 00:27:40,500 --> 00:27:44,380 The other convention is that there are no unique 400 00:27:44,380 --> 00:27:46,400 translations that define a net. 401 00:27:46,400 --> 00:27:49,400 We could take linear combinations of these vectors 402 00:27:49,400 --> 00:27:53,800 and define the same lattice, so we need another rule for 403 00:27:53,800 --> 00:27:55,790 selecting the standard translations-- 404 00:27:55,790 --> 00:27:59,140 again, so that two people can do an x-ray diffraction 405 00:27:59,140 --> 00:28:02,060 experiment and report the results in terms of the same 406 00:28:02,060 --> 00:28:06,150 lattice, and this is a fairly reasonable thing. 407 00:28:06,150 --> 00:28:22,230 This is pick the two shortest translations, and that clearly 408 00:28:22,230 --> 00:28:22,800 makes sense. 409 00:28:22,800 --> 00:28:25,510 There's absolutely nothing at all to commend a cell that has 410 00:28:25,510 --> 00:28:28,830 this as T2 and this as T1 so you get a long, 411 00:28:28,830 --> 00:28:30,982 skinny oblique things. 412 00:28:30,982 --> 00:28:35,460 Your natural inclination would pick the two shortest 413 00:28:35,460 --> 00:28:36,880 translations in the net. 414 00:28:36,880 --> 00:28:38,620 OK, so these are conventions. 415 00:28:38,620 --> 00:28:42,850 This has nothing to do with the nature of the symmetry or 416 00:28:42,850 --> 00:28:46,820 what makes it unique, but just so that people have one 417 00:28:46,820 --> 00:28:49,160 defined way of labeling things. 418 00:28:49,160 --> 00:28:50,044 AUDIENCE: Right. 419 00:28:50,044 --> 00:28:51,370 [INAUDIBLE] my question was actually that-- 420 00:28:51,370 --> 00:28:53,680 PROFESSOR: It was a good answer, even if was not to the 421 00:28:53,680 --> 00:28:55,990 question that you asked. 422 00:28:55,990 --> 00:28:59,372 AUDIENCE: I said the sixfold and the threefold are exactly 423 00:28:59,372 --> 00:29:02,341 the same, but then I realized they are because there is no 424 00:29:02,341 --> 00:29:03,085 crystal that can have threefold 425 00:29:03,085 --> 00:29:05,795 symmetry without sixfold. 426 00:29:05,795 --> 00:29:06,940 PROFESSOR: You were doing fine. 427 00:29:06,940 --> 00:29:10,270 You should have quit just before that last statement. 428 00:29:10,270 --> 00:29:14,950 How can you have a hexagonal lattice that sometimes has 429 00:29:14,950 --> 00:29:17,768 sixfold symmetry and sometimes has threefold symmetry? 430 00:29:17,768 --> 00:29:21,430 Well, let me give you an example for that, and it makes 431 00:29:21,430 --> 00:29:23,015 a very useful point. 432 00:29:26,090 --> 00:29:37,480 Let me draw two hexagonal nets, and in this one I'll put 433 00:29:37,480 --> 00:29:40,890 a threefold axis. 434 00:29:40,890 --> 00:29:47,050 So I'll have one motif here, I'll go 120 degrees away. 435 00:29:47,050 --> 00:29:52,350 Here is a motif here, and I'll go 120 degrees away, and here 436 00:29:52,350 --> 00:29:58,020 is a third motif. 437 00:29:58,020 --> 00:30:00,680 So these three guys form a triangle about 438 00:30:00,680 --> 00:30:02,030 this lattice point. 439 00:30:02,030 --> 00:30:05,740 And we would have about the other lattice points at the 440 00:30:05,740 --> 00:30:17,300 corners of the cell exactly the same triangle of motifs, 441 00:30:17,300 --> 00:30:18,880 and the same thing over here. 442 00:30:26,750 --> 00:30:27,720 So there is a pattern. 443 00:30:27,720 --> 00:30:33,240 It has a hexagonal lattice, and it has a triangle of 444 00:30:33,240 --> 00:30:36,740 objects related by a threefold axis. 445 00:30:36,740 --> 00:30:46,480 And now let me take exactly the same lattice, and now I'll 446 00:30:46,480 --> 00:30:49,910 put in instead a sixfold axis. 447 00:30:49,910 --> 00:30:54,640 And that means I'm going to have a hexagon of objects, and 448 00:30:54,640 --> 00:30:56,120 it'll do something like this. 449 00:31:00,640 --> 00:31:03,430 And I don't want to push my luck and try to draw that 450 00:31:03,430 --> 00:31:07,460 twice, but there would be a hexagon here and a hexagon of 451 00:31:07,460 --> 00:31:09,460 motifs here, another one at this lattice point, and 452 00:31:09,460 --> 00:31:10,860 another one at this lattice point here. 453 00:31:10,860 --> 00:31:14,110 Same lattice, same shape, same dimensions-- 454 00:31:14,110 --> 00:31:15,420 although that's not critical-- 455 00:31:15,420 --> 00:31:18,440 but one of them has only a threefold axis. 456 00:31:18,440 --> 00:31:20,850 One of them has only a sixfold axis in it. 457 00:31:20,850 --> 00:31:21,480 Why? 458 00:31:21,480 --> 00:31:24,290 Because I decided to put a threefold axis in this 459 00:31:24,290 --> 00:31:27,030 pattern, and I decided to put a sixfold axis in this 460 00:31:27,030 --> 00:31:31,390 pattern, but they both end up being contented and happy with 461 00:31:31,390 --> 00:31:33,275 a lattice with the same degree of specialization. 462 00:31:38,230 --> 00:31:40,930 Now we will come as we progressed a little bit 463 00:31:40,930 --> 00:31:42,660 further that we have to go in two dimensions 464 00:31:42,660 --> 00:31:44,200 to the reverse situation. 465 00:31:44,200 --> 00:31:48,100 We have a particular symmetry and it's happy with two 466 00:31:48,100 --> 00:31:50,980 different sorts of lattices with different shapes and 467 00:31:50,980 --> 00:31:53,780 different specializations, and that's 468 00:31:53,780 --> 00:31:56,270 going to come up directly. 469 00:31:56,270 --> 00:31:57,520 Any other questions? 470 00:32:01,050 --> 00:32:02,040 Yeah. 471 00:32:02,040 --> 00:32:07,980 AUDIENCE: In this example, the sixfold [INAUDIBLE] 472 00:32:07,980 --> 00:32:11,445 axis [INAUDIBLE]. 473 00:32:11,445 --> 00:32:12,695 It's just that [INAUDIBLE]. 474 00:32:16,410 --> 00:32:18,680 PROFESSOR: Yes. 475 00:32:18,680 --> 00:32:21,170 You're saying that here hanging at this lattice point 476 00:32:21,170 --> 00:32:23,270 is something that has sixfold symmetry. 477 00:32:23,270 --> 00:32:27,460 Here is something has only threefold symmetry. 478 00:32:27,460 --> 00:32:35,190 The nature of the lattice and the symmetry that is in that 479 00:32:35,190 --> 00:32:39,450 lattice are two inseparable aspects of the pattern. 480 00:32:42,090 --> 00:32:45,840 But often, as we've seen here, there's more than one 481 00:32:45,840 --> 00:32:48,730 possibility for a given lattice. 482 00:32:48,730 --> 00:32:52,750 And as we'll see very shortly, for some other symmetries, for 483 00:32:52,750 --> 00:32:56,470 one given symmetry, there are two kinds of lattices. 484 00:32:56,470 --> 00:33:01,970 But nevertheless, the thing to keep fixed in your mind is 485 00:33:01,970 --> 00:33:04,800 that we call this a hexagonal lattice. 486 00:33:04,800 --> 00:33:05,270 Why? 487 00:33:05,270 --> 00:33:09,120 Because this translation is equal to this one, and they 488 00:33:09,120 --> 00:33:11,940 are exactly 120 degrees apart. 489 00:33:11,940 --> 00:33:15,540 That lattice can have that specialness only if there's 490 00:33:15,540 --> 00:33:19,460 either a threefold or a sixfold axis in it. 491 00:33:19,460 --> 00:33:23,240 So the specialization of a lattice is inseparable from 492 00:33:23,240 --> 00:33:26,900 the symmetry that is in the lattice that demands that 493 00:33:26,900 --> 00:33:28,600 specialization of the lattice. 494 00:33:28,600 --> 00:33:32,940 Conversely, a lattice can have the specialization only if you 495 00:33:32,940 --> 00:33:36,180 place in a symmetry which demands precisely that 496 00:33:36,180 --> 00:33:37,740 specialization. 497 00:33:37,740 --> 00:33:42,450 So if you measure a lattice, and this turns out to be 498 00:33:42,450 --> 00:33:49,050 119.99 degrees and these turn out to be 3.21 angstroms and 499 00:33:49,050 --> 00:33:54,370 3.21 angstroms, that is not a hexagonal lattice, because 500 00:33:54,370 --> 00:33:57,450 there's no symmetry in there that demands that this angle 501 00:33:57,450 --> 00:34:01,100 be 120 degrees. 502 00:34:01,100 --> 00:34:06,180 And that may seem to be an academic fine point, but we'll 503 00:34:06,180 --> 00:34:10,489 see that in due course the properties of a crystal depend 504 00:34:10,489 --> 00:34:13,444 on the symmetry of that crystal. 505 00:34:13,444 --> 00:34:17,239 If the crystal has symmetry, the property also has to have 506 00:34:17,239 --> 00:34:18,449 that symmetry. 507 00:34:18,449 --> 00:34:21,920 And it is the atoms inside the cell which determine the 508 00:34:21,920 --> 00:34:25,580 symmetry of the property, and the properties is one aspect 509 00:34:25,580 --> 00:34:28,620 of the symmetry that goes along with lattice dimensions 510 00:34:28,620 --> 00:34:30,442 and lattice angles. 511 00:34:30,442 --> 00:34:32,310 Is that clear? 512 00:34:32,310 --> 00:34:34,330 So let me say it again, because this is 513 00:34:34,330 --> 00:34:35,679 an important point. 514 00:34:35,679 --> 00:34:39,920 The specialness of a lattice is inseparable from the 515 00:34:39,920 --> 00:34:43,610 symmetry that is existing in that lattice that demands that 516 00:34:43,610 --> 00:34:44,860 specialness. 517 00:34:46,880 --> 00:34:49,949 So if you have a crystal with three orthogonal translations 518 00:34:49,949 --> 00:34:54,050 that is equal in length as you might care to measure, that 519 00:34:54,050 --> 00:34:58,180 crystal is not cubic unless there's symmetry in that 520 00:34:58,180 --> 00:35:02,380 lattice that demands that the edges of the cell conform to 521 00:35:02,380 --> 00:35:03,630 the geometry of a cube. 522 00:35:12,000 --> 00:35:13,250 Any other questions? 523 00:35:18,820 --> 00:35:24,230 Let's then in the time that's remaining look at the other 524 00:35:24,230 --> 00:35:27,540 symmetry element that could be present in a two-dimensional 525 00:35:27,540 --> 00:35:43,570 crystal, and that's the mirror plane. 526 00:35:43,570 --> 00:35:44,910 So here's a mirror plane. 527 00:35:44,910 --> 00:35:48,090 That's the one remaining symmetry element in two 528 00:35:48,090 --> 00:35:55,250 dimensions, and let's ask how we might combine in this space 529 00:35:55,250 --> 00:35:57,700 along with the mirror plane a translation. 530 00:36:00,350 --> 00:36:05,610 If I just pop in a translation, and call this T1, 531 00:36:05,610 --> 00:36:07,820 and for convenience, I'll take the lattice point on the 532 00:36:07,820 --> 00:36:08,870 mirror plane. 533 00:36:08,870 --> 00:36:11,636 Here's another lattice point that sits here. 534 00:36:11,636 --> 00:36:13,720 The mirror plane acts on everything. 535 00:36:13,720 --> 00:36:17,470 It's going to take this lattice point and flip it over 536 00:36:17,470 --> 00:36:20,940 to here, and it's going to take the translation that goes 537 00:36:20,940 --> 00:36:24,280 from the origin lattice point to this lattice point and give 538 00:36:24,280 --> 00:36:26,240 me a T1 prime that sits here. 539 00:36:29,560 --> 00:36:35,310 And now I have two non collinear translations, so 540 00:36:35,310 --> 00:36:40,990 these have defined for me a cell that looks like this. 541 00:36:44,550 --> 00:36:49,620 This is T1, this is T2, and this is some angle alpha 542 00:36:49,620 --> 00:36:52,120 between them. 543 00:36:52,120 --> 00:36:54,400 That's a special lattice. 544 00:36:54,400 --> 00:36:58,240 This is a lattice which has, just as in the hexagonal or 545 00:36:58,240 --> 00:37:01,660 square net, two translations that are 546 00:37:01,660 --> 00:37:03,480 identical in magnitude. 547 00:37:03,480 --> 00:37:03,870 Why? 548 00:37:03,870 --> 00:37:07,650 Because they're repeated by a mirror plane, and the angle 549 00:37:07,650 --> 00:37:11,050 between them-- the angle between T1 and T2-- 550 00:37:11,050 --> 00:37:13,120 is completely general. 551 00:37:16,030 --> 00:37:17,290 It can be anything it likes. 552 00:37:17,290 --> 00:37:20,030 It can close up to a very narrow angle or it can open up 553 00:37:20,030 --> 00:37:21,350 to an almost flat angle. 554 00:37:24,500 --> 00:37:25,750 That's a new kind of lattice. 555 00:37:25,750 --> 00:37:28,450 None of the preceding lattices could make that claim. 556 00:37:33,070 --> 00:37:38,810 Now let me point out that this is for the first time a case 557 00:37:38,810 --> 00:37:44,030 where it would be much to our advantage to not choose a cell 558 00:37:44,030 --> 00:37:46,750 that contains one lattice point. 559 00:37:46,750 --> 00:37:48,650 Let me put some dotted lines in here. 560 00:37:55,850 --> 00:37:58,270 And let me submit-- 561 00:37:58,270 --> 00:38:00,160 these will also be lattice points-- 562 00:38:00,160 --> 00:38:07,010 that I could pick a larger cell that would have this as 563 00:38:07,010 --> 00:38:12,830 T1, it would have this as T2, and the angle between them now 564 00:38:12,830 --> 00:38:17,020 would be identically 90 degrees because of the 565 00:38:17,020 --> 00:38:22,310 geometry that gives us a rhombus here. 566 00:38:22,310 --> 00:38:24,940 This would have two translations that are not 567 00:38:24,940 --> 00:38:31,520 equal in magnitude, and it would have an angle between 568 00:38:31,520 --> 00:38:39,060 these two translations that is identically 90 degrees, but it 569 00:38:39,060 --> 00:38:44,640 is no longer a cell that contains one lattice point. 570 00:38:44,640 --> 00:38:46,975 It now catches a second lattice point 571 00:38:46,975 --> 00:38:49,550 that's in the middle. 572 00:38:49,550 --> 00:38:59,600 So this is a double cell, and it has a rectangular shape. 573 00:38:59,600 --> 00:39:01,245 It's a centered rectangular lattice. 574 00:39:11,450 --> 00:39:15,730 Being a double cell that's redundant has twice the area 575 00:39:15,730 --> 00:39:19,920 that is unique in the pattern, and anything that's hanging up 576 00:39:19,920 --> 00:39:25,040 here is going to be hanging down here, so it has a twofold 577 00:39:25,040 --> 00:39:27,370 redundancy. 578 00:39:27,370 --> 00:39:32,280 But the thing that you get in return for paying the price of 579 00:39:32,280 --> 00:39:37,410 that redundancy is a cell that has a right angle in it. 580 00:39:37,410 --> 00:39:41,300 And as we'll see, we're going to use the edges of the unit 581 00:39:41,300 --> 00:39:44,470 cell as the basis of a coordinate system for 582 00:39:44,470 --> 00:39:48,710 describing what goes on at positions xy within the cell. 583 00:39:48,710 --> 00:39:52,460 And the advantages of an orthogonal coordinate system, 584 00:39:52,460 --> 00:39:56,780 whenever you can take advantage of it, far outweighs 585 00:39:56,780 --> 00:39:59,900 the price of pain-- 586 00:39:59,900 --> 00:40:03,660 twice the area to describe the same pattern. 587 00:40:03,660 --> 00:40:08,190 So this is what is generally taken as the standard cell. 588 00:40:08,190 --> 00:40:09,440 It's a double cell. 589 00:40:13,890 --> 00:40:16,880 But I think you're used to that sort of compromise, 590 00:40:16,880 --> 00:40:20,270 because you've all heard of face centered cubic lattices. 591 00:40:20,270 --> 00:40:22,960 The primitive cell in a face centered cubic lattice is a 592 00:40:22,960 --> 00:40:25,100 rhombohedron. 593 00:40:25,100 --> 00:40:29,600 But, oh, that Cartesian coordinate system is so great 594 00:40:29,600 --> 00:40:32,140 to use rather than something that has an 595 00:40:32,140 --> 00:40:34,680 oblique coordinate system. 596 00:40:34,680 --> 00:40:37,450 So is this is a definition of convenience. 597 00:40:37,450 --> 00:40:39,830 Notice the curious duality-- 598 00:40:39,830 --> 00:40:42,890 either special relation between the translations angle 599 00:40:42,890 --> 00:40:47,350 general or special relation between the translations, an 600 00:40:47,350 --> 00:40:48,600 angle special. 601 00:40:50,700 --> 00:40:55,020 General translation, special angle, relation between the 602 00:40:55,020 --> 00:40:57,200 translation, general angle-- 603 00:40:57,200 --> 00:40:58,980 so, it's a curious sort of duality. 604 00:40:58,980 --> 00:41:01,900 You can have one but not the other or vice versa. 605 00:41:07,050 --> 00:41:13,380 So this is a fourth sort of lattice. 606 00:41:13,380 --> 00:41:19,630 Number one, number two, number three, and now we have number 607 00:41:19,630 --> 00:41:22,400 four, which is a centered rectangular lattice. 608 00:41:36,430 --> 00:41:46,460 And this particular lattice has T1 not equal to T2 in 609 00:41:46,460 --> 00:41:53,220 magnitude, but it has the angle between T1 and T2 610 00:41:53,220 --> 00:41:56,305 exactly 90 degrees, and it's a double cell. 611 00:42:02,590 --> 00:42:03,840 It's centered. 612 00:42:13,440 --> 00:42:14,690 Are we done? 613 00:42:18,460 --> 00:42:21,280 The reason I asked that silly rhetorical question is that 614 00:42:21,280 --> 00:42:25,490 obviously I suspect that we're not done, and what 615 00:42:25,490 --> 00:42:28,380 else might we do? 616 00:42:28,380 --> 00:42:31,080 We got our centered rectangular net by starting 617 00:42:31,080 --> 00:42:33,460 with a translation-- 618 00:42:33,460 --> 00:42:36,320 starting with an mirror plane, really-- 619 00:42:36,320 --> 00:42:40,430 and then we added a general translation, and that 620 00:42:40,430 --> 00:42:43,710 reflected it across and gave us a diamond shaped net which 621 00:42:43,710 --> 00:42:47,630 we could define as a rectangular double shell. 622 00:42:47,630 --> 00:42:49,542 Does that always happen? 623 00:42:49,542 --> 00:42:53,280 Do I always get that oblique diamond shaped cell? 624 00:42:58,260 --> 00:42:59,660 No. 625 00:42:59,660 --> 00:43:06,690 Suppose I put in my first translation deliberately in a 626 00:43:06,690 --> 00:43:10,150 fashion such that it was at exactly right angles to the 627 00:43:10,150 --> 00:43:13,080 mirror plane. 628 00:43:13,080 --> 00:43:17,090 That mirror plane will then reflect the translation and 629 00:43:17,090 --> 00:43:21,050 change its direction, and now I have generated a 630 00:43:21,050 --> 00:43:23,160 one-dimensional lattice row with 631 00:43:23,160 --> 00:43:26,630 translational periodicity T1. 632 00:43:26,630 --> 00:43:29,020 And I've got a translation that's exactly perpendicular 633 00:43:29,020 --> 00:43:30,270 to the mirror plane. 634 00:43:33,720 --> 00:43:37,460 How do I now make a space lattice, a two-dimensional 635 00:43:37,460 --> 00:43:39,150 space lattice? 636 00:43:39,150 --> 00:43:44,390 And the answer is very carefully. 637 00:43:44,390 --> 00:43:49,575 Suppose I throw in a second translation T2 and the mirror 638 00:43:49,575 --> 00:43:52,290 plane reflects it across to here. 639 00:43:52,290 --> 00:43:55,760 This interval between lattice points up here is totally in 640 00:43:55,760 --> 00:43:57,710 commensurate with the first translation, 641 00:43:57,710 --> 00:44:00,280 and that won't work. 642 00:44:00,280 --> 00:44:03,670 It's violated my initial choice of the translational 643 00:44:03,670 --> 00:44:10,540 periodicity unless I do one of two things, and let's try them 644 00:44:10,540 --> 00:44:13,180 both and dispose of this quickly-- 645 00:44:13,180 --> 00:44:15,050 this is straight away. 646 00:44:15,050 --> 00:44:18,120 Here's my one-dimensional lattice row. 647 00:44:18,120 --> 00:44:21,670 I do not violate this periodicity 648 00:44:21,670 --> 00:44:23,650 T1 if I do two things. 649 00:44:23,650 --> 00:44:29,540 I could pick T2 so that it fell exactly along the mirror 650 00:44:29,540 --> 00:44:34,840 line, and that's going to generate for me a new type of 651 00:44:34,840 --> 00:44:38,880 lattice in which this is 90 degrees and these two 652 00:44:38,880 --> 00:44:43,140 translations are unequal in length. 653 00:44:43,140 --> 00:44:47,900 So this is a lattice that has a rectangular shape. 654 00:44:47,900 --> 00:45:02,870 It's a primitive rectangular cell, and it has T1 not equal 655 00:45:02,870 --> 00:45:12,760 to T2 in magnitude, and it has T1 and T2 656 00:45:12,760 --> 00:45:14,480 exactly 90 degrees apart. 657 00:45:17,070 --> 00:45:19,950 The second choice that would not result in any 658 00:45:19,950 --> 00:45:26,540 contradiction would be to have this as T1, and then pick T2 659 00:45:26,540 --> 00:45:33,240 very carefully so that it spanned the mirror line with 660 00:45:33,240 --> 00:45:36,380 one half of T1 exactly up here and one half 661 00:45:36,380 --> 00:45:38,470 of T1 exactly here. 662 00:45:38,470 --> 00:45:42,590 And this would be compatible with the separation T1 down at 663 00:45:42,590 --> 00:45:44,760 the start of this translation. 664 00:45:44,760 --> 00:45:47,980 I think you can see that what this is going to give me is 665 00:45:47,980 --> 00:45:51,480 the centered rectangular net right back again, so this is 666 00:45:51,480 --> 00:45:53,300 the centered rectangular net-- 667 00:46:02,890 --> 00:46:04,140 nothing new. 668 00:46:09,420 --> 00:46:13,740 But we did pick up one additional two-dimensional 669 00:46:13,740 --> 00:46:16,420 lattice of distinct character. 670 00:46:16,420 --> 00:46:29,020 Number 5 is a primitive rectangular network, and it 671 00:46:29,020 --> 00:46:36,810 has the characteristics T1 is not equal to T2 in magnitude. 672 00:46:36,810 --> 00:46:40,510 Just as in the centered rectangular net, T1 is at 673 00:46:40,510 --> 00:46:44,490 exactly 90 degrees to T2, but this is a primitive cell. 674 00:46:50,450 --> 00:46:52,456 And that's it. 675 00:46:52,456 --> 00:46:56,280 We really set up the ground rules for the geometry of a 676 00:46:56,280 --> 00:46:59,170 periodic two-dimensional space. 677 00:46:59,170 --> 00:47:04,190 There are five kinds of rotation axes-- 678 00:47:04,190 --> 00:47:06,420 one, two, three, four, and six. 679 00:47:06,420 --> 00:47:12,030 Each one requires one or more of the specialized 680 00:47:12,030 --> 00:47:14,040 two-dimensional lattices. 681 00:47:14,040 --> 00:47:17,970 We have a case where interestingly two different 682 00:47:17,970 --> 00:47:21,740 symmetries are compatible with a lattice of the same 683 00:47:21,740 --> 00:47:22,720 specialization. 684 00:47:22,720 --> 00:47:24,600 In the case of the [? hexagon ?] on that, either 685 00:47:24,600 --> 00:47:28,130 three or sixfold symmetry could require that. 686 00:47:28,130 --> 00:47:31,730 In the case of the mirror plane we have one symmetry 687 00:47:31,730 --> 00:47:34,040 element, M, that can fit into two 688 00:47:34,040 --> 00:47:37,620 different kinds of lattices. 689 00:47:37,620 --> 00:47:39,630 So in one case, the same lattice can take two different 690 00:47:39,630 --> 00:47:40,310 symmetries. 691 00:47:40,310 --> 00:47:43,850 In this case, two different lattices can accommodate the 692 00:47:43,850 --> 00:47:45,100 same symmetry. 693 00:47:47,070 --> 00:47:51,430 So that's the story for two dimensions, and we have just 694 00:47:51,430 --> 00:47:53,740 one final thing to do. 695 00:47:53,740 --> 00:48:11,550 That is to add to the lattices that we have found, and there 696 00:48:11,550 --> 00:48:15,850 are five of them, and add to the lattice point the 697 00:48:15,850 --> 00:48:21,020 symmetries that we have found require them. 698 00:48:21,020 --> 00:48:28,400 And there are a limited number of these-- one, two, three, 699 00:48:28,400 --> 00:48:32,760 four, or sixfold rotational symmetry in a mirror plane. 700 00:48:32,760 --> 00:48:39,640 And when we have finished these additions, we will end 701 00:48:39,640 --> 00:48:44,110 up with a combination of lattice and symmetry, which is 702 00:48:44,110 --> 00:48:47,160 something that is called a plane group-- 703 00:48:51,590 --> 00:48:57,840 group because the operations that are present in the space 704 00:48:57,840 --> 00:49:03,370 follow the requirements of the mathematical entity called a 705 00:49:03,370 --> 00:49:08,330 group, plane because this is a distribution of symmetry 706 00:49:08,330 --> 00:49:12,000 elements throughout a plane and not just fixed at one 707 00:49:12,000 --> 00:49:15,200 particular point. 708 00:49:15,200 --> 00:49:18,690 Let me finish with some general observation, and we 709 00:49:18,690 --> 00:49:23,110 will obtain some of these rules later on. 710 00:49:23,110 --> 00:49:25,320 Suppose I take a pristine space-- 711 00:49:25,320 --> 00:49:27,880 and this blackboard is no longer pristine-- 712 00:49:27,880 --> 00:49:31,690 and I put in the space a first operation, and 713 00:49:31,690 --> 00:49:32,710 there's a motif in there. 714 00:49:32,710 --> 00:49:37,260 This first operation moves around this first motif and 715 00:49:37,260 --> 00:49:40,910 gives me a second one, number two. 716 00:49:40,910 --> 00:49:42,850 Then I say let me put in another operation. 717 00:49:42,850 --> 00:49:45,270 I'd like to combine these things and see how many 718 00:49:45,270 --> 00:49:47,360 different combinations I have, so I put in 719 00:49:47,360 --> 00:49:49,090 operation number two. 720 00:49:49,090 --> 00:49:53,660 Operation number two will take the second object and repeat 721 00:49:53,660 --> 00:49:54,970 it to a third object. 722 00:49:58,700 --> 00:50:01,680 Number two is identical to number one, so this gives me a 723 00:50:01,680 --> 00:50:03,540 third object reproduced from the second by 724 00:50:03,540 --> 00:50:04,970 operation number two. 725 00:50:04,970 --> 00:50:07,920 Now I have a space, and sitting in it are two 726 00:50:07,920 --> 00:50:11,520 different operations and three different motifs. 727 00:50:11,520 --> 00:50:15,260 Motif number one and motif number three are the same darn 728 00:50:15,260 --> 00:50:18,490 thing because they've been repeated by symmetry steps, so 729 00:50:18,490 --> 00:50:26,860 there must automatically be some third transformation that 730 00:50:26,860 --> 00:50:30,740 is equal to the combined effect of going from one to 731 00:50:30,740 --> 00:50:32,300 two and then from two to three-- 732 00:50:32,300 --> 00:50:35,670 going from one to three directly. 733 00:50:35,670 --> 00:50:40,040 So this is another truth about these symmetries. 734 00:50:40,040 --> 00:50:45,790 Whenever you take two operations and combine them in 735 00:50:45,790 --> 00:50:52,060 a space, the net effect of those two operations is equal 736 00:50:52,060 --> 00:50:55,250 to a third operation. 737 00:50:55,250 --> 00:50:58,980 So a question we're going to ask all along the way until we 738 00:50:58,980 --> 00:51:00,480 are at the end of the month-- 739 00:51:00,480 --> 00:51:03,070 if you take a translation and combine it with a mirror 740 00:51:03,070 --> 00:51:07,270 plane, what new operation has to arise? 741 00:51:07,270 --> 00:51:09,470 If you take two rotation operations and put them 742 00:51:09,470 --> 00:51:13,370 together, what third net operation has to arrive? 743 00:51:13,370 --> 00:51:16,760 If we have some general rules, then we can automatically say, 744 00:51:16,760 --> 00:51:18,620 OK, I'm going to take a square lattice and I'm going to put 745 00:51:18,620 --> 00:51:19,800 in a fourfold axis. 746 00:51:19,800 --> 00:51:22,750 What else is going to pop up elsewhere within the cell? 747 00:51:22,750 --> 00:51:24,990 We'll be able to do this systematically but fairly 748 00:51:24,990 --> 00:51:27,240 automatically. 749 00:51:27,240 --> 00:51:29,530 So that's where we're going from here. 750 00:51:29,530 --> 00:51:33,690 When we're done, we will have derived systematically and 751 00:51:33,690 --> 00:51:39,690 rigorously the sorts of symmetries that can combine 752 00:51:39,690 --> 00:51:43,200 symmetries such as rotation and reflection with a lattice, 753 00:51:43,200 --> 00:51:46,340 and we will know completely the different sorts of 754 00:51:46,340 --> 00:51:49,130 patterns that exist around us in two dimensions-- 755 00:51:49,130 --> 00:51:51,510 in floor tiles, brick work, wrapping 756 00:51:51,510 --> 00:51:54,130 paper, and plaid shirts. 757 00:51:57,190 --> 00:52:01,170 We'll pick up at this point on Thursday. 758 00:52:01,170 --> 00:52:05,680 Let me caution you we are going into territory that is 759 00:52:05,680 --> 00:52:07,220 not covered in Berger's book. 760 00:52:07,220 --> 00:52:09,300 I've just passed it out to you. 761 00:52:09,300 --> 00:52:12,720 What we've said in the early parts of the term are in 762 00:52:12,720 --> 00:52:15,290 there, but we're going to do things in a slightly different 763 00:52:15,290 --> 00:52:17,550 way and then return to his text later on.