1 00:00:10,538 --> 00:00:14,420 PROFESSOR: I wanted to make a few comments on the third 2 00:00:14,420 --> 00:00:17,530 problem set, which was a funny puzzle. 3 00:00:17,530 --> 00:00:20,320 And it was intended to be entertaining and get you 4 00:00:20,320 --> 00:00:23,320 thinking about things related to symmetry 5 00:00:23,320 --> 00:00:25,250 in an amusing context. 6 00:00:25,250 --> 00:00:26,640 And there were two problems. 7 00:00:26,640 --> 00:00:32,030 The first problem had a platter on which there were 8 00:00:32,030 --> 00:00:37,650 arranged cherries with either two or three berries to a 9 00:00:37,650 --> 00:00:40,940 spray, and apples and pears. 10 00:00:40,940 --> 00:00:43,300 And they were either black or white, which didn't make then 11 00:00:43,300 --> 00:00:45,130 particularly appetizing. 12 00:00:45,130 --> 00:00:51,140 But in any case, there were three platters and the fruits 13 00:00:51,140 --> 00:00:53,020 marched around on the platter. 14 00:00:53,020 --> 00:00:57,760 And the puzzle was, what would be the fourth 15 00:00:57,760 --> 00:01:00,280 platter in the sequence? 16 00:01:00,280 --> 00:01:04,090 This didn't involve symmetry in any sense of the word. 17 00:01:04,090 --> 00:01:09,120 But yet, what was going on from picture to picture was a 18 00:01:09,120 --> 00:01:13,410 mapping of the motifs from one location to another. 19 00:01:13,410 --> 00:01:18,430 Not just rotation or translation or anything nice 20 00:01:18,430 --> 00:01:21,870 in the crystallographic, but still they were being moved 21 00:01:21,870 --> 00:01:24,010 from one location to another. 22 00:01:24,010 --> 00:01:27,300 And just about everybody got the right 23 00:01:27,300 --> 00:01:31,420 answer just by deduction. 24 00:01:31,420 --> 00:01:34,430 But the reason I liked that problem, and the reason I gave 25 00:01:34,430 --> 00:01:38,030 it to you, is that it introduced another sort of 26 00:01:38,030 --> 00:01:40,660 transformation. 27 00:01:40,660 --> 00:01:44,000 One of the pairs of fruits not only change locations, but 28 00:01:44,000 --> 00:01:47,150 they switched from black to white as you went from one 29 00:01:47,150 --> 00:01:49,260 position to the other. 30 00:01:49,260 --> 00:01:50,880 And that is another sort of 31 00:01:50,880 --> 00:01:53,250 transformation we can introduce. 32 00:01:53,250 --> 00:01:56,250 And this is something we could call color symmetry. 33 00:01:56,250 --> 00:02:01,230 And I would observe that this is something that you are 34 00:02:01,230 --> 00:02:06,830 already very familiar with in patterns, in particular, in 35 00:02:06,830 --> 00:02:13,760 the pattern that is represented by a checkerboard 36 00:02:13,760 --> 00:02:18,140 where you have black squares and white squares. 37 00:02:18,140 --> 00:02:23,450 This location here is a bona fide four-fold axis, as is 38 00:02:23,450 --> 00:02:26,010 this location here. 39 00:02:26,010 --> 00:02:30,670 But there other locations, such as the corners of the 40 00:02:30,670 --> 00:02:35,180 black and white squares, where, indeed, there is a 90 41 00:02:35,180 --> 00:02:38,390 degree rotation that takes this square and transforms it 42 00:02:38,390 --> 00:02:39,360 into this square. 43 00:02:39,360 --> 00:02:41,680 But in doing that transformation, you change it 44 00:02:41,680 --> 00:02:45,360 from a black one to a white one, and then from a white one 45 00:02:45,360 --> 00:02:48,750 to a black one again, back to a white one. 46 00:02:48,750 --> 00:02:52,962 And that is a color symmetry. 47 00:02:52,962 --> 00:02:55,880 You could make it red and green if you like to be a 48 00:02:55,880 --> 00:02:57,280 little more attractive. 49 00:02:57,280 --> 00:03:01,190 Similarly, there are mirror lines like this that are true 50 00:03:01,190 --> 00:03:02,240 mirror lines. 51 00:03:02,240 --> 00:03:04,030 The pattern is left invariant. 52 00:03:04,030 --> 00:03:07,140 There's another locus such as this one here where when you 53 00:03:07,140 --> 00:03:11,050 reflect you go from a black one to white one and then back 54 00:03:11,050 --> 00:03:14,600 to a black one when you do the operation twice. 55 00:03:14,600 --> 00:03:17,230 So there are black-white mirror planes as well as 56 00:03:17,230 --> 00:03:18,830 regular mirror planes. 57 00:03:18,830 --> 00:03:22,160 There are two-fold axes but all of the two-fold axes are 58 00:03:22,160 --> 00:03:26,390 black-white axes, white to black, black back to white. 59 00:03:26,390 --> 00:03:30,070 So that's a new sort of transformation that we can 60 00:03:30,070 --> 00:03:32,540 develop if we wanted to, switching the color. 61 00:03:35,370 --> 00:03:37,910 Does this is have any utility in the physical sciences? 62 00:03:37,910 --> 00:03:40,510 Be nice for wallpaper designers, but does it have 63 00:03:40,510 --> 00:03:42,360 utility in physical sciences? 64 00:03:42,360 --> 00:03:43,990 And the answer is yes. 65 00:03:43,990 --> 00:03:47,760 Because what is going on here is that there is a binary 66 00:03:47,760 --> 00:03:50,500 character assigned to each of the motifs. 67 00:03:50,500 --> 00:03:52,990 So any time the atom or whatever you have, the 68 00:03:52,990 --> 00:03:57,240 molecule, has some sort of binary characteristic that it 69 00:03:57,240 --> 00:04:00,840 can exist in two distinct states, the arrangement of 70 00:04:00,840 --> 00:04:05,080 that sort of motif requires a black-white symmetry. 71 00:04:05,080 --> 00:04:09,080 And one example in crystals that requires this sort of 72 00:04:09,080 --> 00:04:12,610 symmetry is the case of a magnetic structure-- 73 00:04:12,610 --> 00:04:15,480 and I think I mentioned this last time-- where the magnetic 74 00:04:15,480 --> 00:04:17,760 moment can either point up or down. 75 00:04:17,760 --> 00:04:20,500 Well, saying that you have an atom with spin up and an atom 76 00:04:20,500 --> 00:04:23,470 with spin down is the same thing as saying, in terms of 77 00:04:23,470 --> 00:04:27,140 transformations, having a black atom or a white atom. 78 00:04:27,140 --> 00:04:33,270 It's a binary character that switches from atom to atom. 79 00:04:33,270 --> 00:04:36,314 Could you have three color symmetries? 80 00:04:36,314 --> 00:04:36,746 Yeah. 81 00:04:36,746 --> 00:04:37,830 You can. 82 00:04:37,830 --> 00:04:44,020 There is a Dutch artist named Maurits Escher who spent much 83 00:04:44,020 --> 00:04:48,280 of his, career deducing patterns, and did this in a 84 00:04:48,280 --> 00:04:50,080 purely intuitive way. 85 00:04:50,080 --> 00:04:52,100 I mean you look at these patterns and say, wow, that 86 00:04:52,100 --> 00:04:54,470 guy is some crystallographer. 87 00:04:54,470 --> 00:04:58,310 But I heard him, while he was still alive, speak twice. 88 00:04:58,310 --> 00:05:01,700 And he describes it in metaphysical terms. 89 00:05:01,700 --> 00:05:06,330 This is the twoness merging with the fourness, and just 90 00:05:06,330 --> 00:05:08,580 completely intuitive. 91 00:05:08,580 --> 00:05:09,790 But his patterns are gorgeous. 92 00:05:09,790 --> 00:05:12,650 And he has a number of very nice ones that are three-color 93 00:05:12,650 --> 00:05:13,790 symmetries. 94 00:05:13,790 --> 00:05:16,410 And maybe later on in the term, I'll bring in a 95 00:05:16,410 --> 00:05:18,520 projector and show you examples of some of these. 96 00:05:18,520 --> 00:05:19,770 They're very entertaining. 97 00:05:22,280 --> 00:05:22,520 All right. 98 00:05:22,520 --> 00:05:25,420 So the purpose of that first puzzle was to introduce you to 99 00:05:25,420 --> 00:05:26,670 black-white symmetries. 100 00:05:28,990 --> 00:05:32,390 Then the second puzzle on the sheet was a pretty dumb thing. 101 00:05:32,390 --> 00:05:36,310 There was a stack of blocks and you said how many-- 102 00:05:36,310 --> 00:05:40,070 I asked how many blocks are there in the stack? 103 00:05:40,070 --> 00:05:42,250 And you could count them up, 1, 2, 3, 4. 104 00:05:42,250 --> 00:05:44,790 And If you did it right, you came out with 32. 105 00:05:44,790 --> 00:05:46,570 So that was pretty dumb. 106 00:05:46,570 --> 00:05:51,090 Well, this was intended as an example of how one can use 107 00:05:51,090 --> 00:05:56,610 symmetry in the solution of a physical problem. 108 00:05:56,610 --> 00:05:59,140 Let's suppose that you were a cowboy and you had to go out 109 00:05:59,140 --> 00:06:02,230 and buy horseshoes for a herd of horses. 110 00:06:02,230 --> 00:06:04,660 Would you count the number of feet? 111 00:06:04,660 --> 00:06:04,960 No. 112 00:06:04,960 --> 00:06:08,924 You'd count the number of horses and multiply by four. 113 00:06:08,924 --> 00:06:09,750 OK. 114 00:06:09,750 --> 00:06:12,061 So there you'd be using symmetry. 115 00:06:12,061 --> 00:06:15,826 Now The point of this problem really was-- 116 00:06:15,826 --> 00:06:17,535 You could have had this one. 117 00:06:17,535 --> 00:06:20,600 We have just the exact number seats. 118 00:06:20,600 --> 00:06:27,092 The point of this problem was that you have to, in assigning 119 00:06:27,092 --> 00:06:31,140 a symmetry to a system, usually make physical 120 00:06:31,140 --> 00:06:32,290 assumptions. 121 00:06:32,290 --> 00:06:35,470 When you do a derivation, as we've been doing for the last 122 00:06:35,470 --> 00:06:38,490 several days, we say this is a four-fold access, and there's 123 00:06:38,490 --> 00:06:40,410 no ambiguity whatsoever. 124 00:06:40,410 --> 00:06:43,110 Because it's my ball game, and I want it to be a four-fold 125 00:06:43,110 --> 00:06:44,720 axis, 126 00:06:44,720 --> 00:06:47,480 But when you're given a pattern or given a crystal and 127 00:06:47,480 --> 00:06:50,500 you have to assign a symmetry to that crystal, you 128 00:06:50,500 --> 00:06:53,840 invariably have to make assumptions. 129 00:06:53,840 --> 00:06:56,410 The person buying the horseshoes, for example, has 130 00:06:56,410 --> 00:07:00,390 to assume that five- and six-legged horses are pretty 131 00:07:00,390 --> 00:07:07,350 rare articles and that a horse with only three or two legs is 132 00:07:07,350 --> 00:07:09,905 going to be of little utility to the rancher. 133 00:07:09,905 --> 00:07:12,680 It Would probably have a limited existence at the 134 00:07:12,680 --> 00:07:13,960 ranchers expense. 135 00:07:13,960 --> 00:07:19,240 So you're assuming that all horse have four legs. 136 00:07:19,240 --> 00:07:23,110 In looking at a crystal, though, things are not that 137 00:07:23,110 --> 00:07:24,460 easy to decide. 138 00:07:24,460 --> 00:07:28,140 And let me give you a few really practical examples. 139 00:07:28,140 --> 00:07:31,490 A lot of mineral crystals grow in fissures in the rock. 140 00:07:31,490 --> 00:07:34,970 And let's suppose we've got a crack, and bubbling through 141 00:07:34,970 --> 00:07:39,050 this crack is a solution that contains silica. 142 00:07:39,050 --> 00:07:41,690 And as the solution cools off, it will deposit 143 00:07:41,690 --> 00:07:43,140 a crystal of quartz. 144 00:07:43,140 --> 00:07:47,010 And this crystal of quartz would probably grow looking 145 00:07:47,010 --> 00:07:49,370 something like this. 146 00:07:49,370 --> 00:07:51,440 And you would knock the crystal out of the rock. 147 00:07:51,440 --> 00:07:54,075 And you would say, what is the symmetry of this crystal? 148 00:07:56,890 --> 00:07:59,890 It doesn't appear to be anything hexagonal about it. 149 00:08:02,420 --> 00:08:06,990 But what's happened is this crystal if it had developed in 150 00:08:06,990 --> 00:08:11,660 a uniform environment would have had a hexagonal shape. 151 00:08:11,660 --> 00:08:15,040 But because the nutrients are flowing in from one direction, 152 00:08:15,040 --> 00:08:17,990 it tends to elongate in that direction. 153 00:08:17,990 --> 00:08:22,640 So how could you assign the symmetry to that crystal that 154 00:08:22,640 --> 00:08:24,840 you dug out of the rock? 155 00:08:24,840 --> 00:08:28,330 Unfortunately, crystals do not come with a legend on the 156 00:08:28,330 --> 00:08:37,409 bottom that says quartz SIO2 space group P3 subscript 121, 157 00:08:37,409 --> 00:08:38,960 made in USA. 158 00:08:38,960 --> 00:08:42,860 You have to yourself assign the symmetry. 159 00:08:42,860 --> 00:08:45,500 And what you would do is you would look at this face and 160 00:08:45,500 --> 00:08:48,050 this face and you say, gee, these angles, when I measure 161 00:08:48,050 --> 00:08:51,720 them, all turn out to be 120 degrees. 162 00:08:51,720 --> 00:08:54,610 This face has a luster on it that looks like the 163 00:08:54,610 --> 00:08:56,190 luster on this face. 164 00:08:56,190 --> 00:08:59,360 This face has little edge pits on it, and they point in the 165 00:08:59,360 --> 00:09:02,820 same way as the edge pits on this face. 166 00:09:02,820 --> 00:09:06,190 I can measure the electrical conductivity and find that it 167 00:09:06,190 --> 00:09:09,830 varies in a fashion consistent with a hexagonal crystal. 168 00:09:09,830 --> 00:09:13,390 And you would do any physical test that you decided you 169 00:09:13,390 --> 00:09:14,430 would need to make. 170 00:09:14,430 --> 00:09:17,650 And then when you were all done, you would say everything 171 00:09:17,650 --> 00:09:20,810 that I can measure about this crystal other than it's shape 172 00:09:20,810 --> 00:09:28,060 is invariant to a 120 degree rotation. 173 00:09:28,060 --> 00:09:34,100 Quartz has a hexagonal shape very often, but actually only 174 00:09:34,100 --> 00:09:38,120 this face, this face, and this face are symmetrically 175 00:09:38,120 --> 00:09:38,760 equivalent. 176 00:09:38,760 --> 00:09:41,350 And there's only a three-fold axis in there. 177 00:09:41,350 --> 00:09:44,550 And if you look very carefully at this crystal, you could see 178 00:09:44,550 --> 00:09:46,130 some difference there, too. 179 00:09:46,130 --> 00:09:50,085 The neighboring faces very often have striations, very 180 00:09:50,085 --> 00:09:53,210 faint striations on one face, no striations on the 181 00:09:53,210 --> 00:09:54,890 neighboring face. 182 00:09:54,890 --> 00:09:56,540 So you find out all that can. 183 00:09:56,540 --> 00:09:59,550 You have to make some assumptions, and then on that 184 00:09:59,550 --> 00:10:02,860 basis decide the symmetry. 185 00:10:02,860 --> 00:10:03,920 Let me give you another example. 186 00:10:03,920 --> 00:10:04,960 This is a real one. 187 00:10:04,960 --> 00:10:08,210 And one of the things that it's fun to do is to grow 188 00:10:08,210 --> 00:10:09,780 crystals from solution. 189 00:10:09,780 --> 00:10:13,040 The Science Museum has little kits that you can buy that 190 00:10:13,040 --> 00:10:16,150 have some powder that you could heat up in water and 191 00:10:16,150 --> 00:10:20,420 then put it in a beaker and put a string in there, maybe 192 00:10:20,420 --> 00:10:23,480 with a little crumb of the stuff that you dissolved. 193 00:10:23,480 --> 00:10:26,340 And one of the crystals that is nice for 194 00:10:26,340 --> 00:10:30,410 this purpose is alum. 195 00:10:30,410 --> 00:10:31,790 It's a cubic crystal. 196 00:10:31,790 --> 00:10:36,810 And if you let the stuff set there, it will form a nice, 197 00:10:36,810 --> 00:10:39,063 lovely equiaxed cubed. 198 00:10:39,063 --> 00:10:43,190 Science Museum usually puts a pinch of another salt in there 199 00:10:43,190 --> 00:10:46,330 so the crystal is not only pretty and shiny, it's purple 200 00:10:46,330 --> 00:10:48,725 or it's the orange. 201 00:10:48,725 --> 00:10:52,830 Now, unless you attach this little seed crystal carefully, 202 00:10:52,830 --> 00:10:56,230 there would be a good chance that it would fall off and 203 00:10:56,230 --> 00:10:57,850 land on the bottom of the beaker. 204 00:10:57,850 --> 00:11:01,180 When that happened, your crystal would not have the 205 00:11:01,180 --> 00:11:03,670 shape of a cube. 206 00:11:03,670 --> 00:11:08,060 It would have a shape in which these two top edges had length 207 00:11:08,060 --> 00:11:13,970 L and these faces on the side were exactly half that length. 208 00:11:13,970 --> 00:11:16,560 And what's happening is that this crystal is growing in an 209 00:11:16,560 --> 00:11:18,335 unconstrained environment. 210 00:11:18,335 --> 00:11:21,640 The nutrient is coming on and crystallizing from all 211 00:11:21,640 --> 00:11:22,820 directions. 212 00:11:22,820 --> 00:11:27,830 In this case, nutrient can't get in from the directions 213 00:11:27,830 --> 00:11:29,270 that are below the crystal. 214 00:11:29,270 --> 00:11:32,780 So the growth in that direction is eliminated, and 215 00:11:32,780 --> 00:11:35,100 this height is exactly half that at 216 00:11:35,100 --> 00:11:37,880 the edge of the crystal. 217 00:11:37,880 --> 00:11:41,480 So that's sort of a very specialized case of a crystal 218 00:11:41,480 --> 00:11:43,225 growing in a constrained environment. 219 00:11:47,160 --> 00:11:50,110 I'm digressing now, but let me make a few more remarks. 220 00:11:50,110 --> 00:11:52,485 The shape that a crystal has-- 221 00:11:55,930 --> 00:11:59,170 and the shape is in crystallography designated by 222 00:11:59,170 --> 00:12:00,460 a special term. 223 00:12:00,460 --> 00:12:01,710 It's called habit. 224 00:12:06,450 --> 00:12:10,960 Crystals that undergo plastic deformation never die, they 225 00:12:10,960 --> 00:12:12,280 just develop bad habits. 226 00:12:15,020 --> 00:12:15,590 Oh, come on. 227 00:12:15,590 --> 00:12:18,070 This is a tough crowd. 228 00:12:18,070 --> 00:12:19,510 I thought that was pretty funny. 229 00:12:19,510 --> 00:12:20,810 Anyway. 230 00:12:20,810 --> 00:12:25,420 The habit that a crystal has can depend very much on 231 00:12:25,420 --> 00:12:29,730 impurities in the solution or melt from which it grows. 232 00:12:29,730 --> 00:12:33,340 And let me suppose I start with a little crystal that 233 00:12:33,340 --> 00:12:35,650 looks like this. 234 00:12:35,650 --> 00:12:39,640 And there are two different kinds of faces on there. 235 00:12:39,640 --> 00:12:42,540 And let's suppose one of them grows rapidly and one of them 236 00:12:42,540 --> 00:12:44,060 grows slowly. 237 00:12:44,060 --> 00:12:48,050 Which face do you think would be the one that eventually 238 00:12:48,050 --> 00:12:51,000 forms the boundaries to the crystal volume? 239 00:12:51,000 --> 00:12:54,714 The fast-growing face or the slow-growing face? 240 00:12:54,714 --> 00:12:55,670 AUDIENCE: Fast-growing face. 241 00:12:55,670 --> 00:12:58,640 PROFESSOR: Fast, slow are both possibilities. 242 00:12:58,640 --> 00:13:00,320 Let's do a time lapse experiment. 243 00:13:00,320 --> 00:13:02,840 Let's suppose that this is the fast-growing face. 244 00:13:02,840 --> 00:13:06,070 And, after a certain amount of time these faces would have 245 00:13:06,070 --> 00:13:09,260 all advanced to this location. 246 00:13:09,260 --> 00:13:10,820 And this is the slow-growing face. 247 00:13:10,820 --> 00:13:14,210 And after the same amount of time, it will have advanced to 248 00:13:14,210 --> 00:13:16,550 this location. 249 00:13:16,550 --> 00:13:18,650 After the next time increment, this one would 250 00:13:18,650 --> 00:13:20,100 have grown up to here. 251 00:13:20,100 --> 00:13:22,270 This would only have grown that far. 252 00:13:22,270 --> 00:13:25,620 And you could see the only thing that's gonna be left is 253 00:13:25,620 --> 00:13:27,350 the slow-growing face. 254 00:13:27,350 --> 00:13:29,480 So it's sort of counter intuitive. 255 00:13:29,480 --> 00:13:32,740 The faces that survive to bound the crystal as rational 256 00:13:32,740 --> 00:13:35,430 planes are the slow-growing faces, not the 257 00:13:35,430 --> 00:13:36,680 rapidly growing faces. 258 00:13:39,050 --> 00:13:42,670 This is of great use to the people who want to control 259 00:13:42,670 --> 00:13:45,330 crystal shape. 260 00:13:45,330 --> 00:13:51,020 And this is very often in very every day, ordinary contexts. 261 00:13:51,020 --> 00:13:53,890 The people who grow table salt, for example, want to 262 00:13:53,890 --> 00:13:57,360 grow a crystal that has nice pointy edges so when it comes 263 00:13:57,360 --> 00:13:59,880 down on your tomato, it doesn't bounce off. 264 00:13:59,880 --> 00:14:02,110 It sticks to it. 265 00:14:02,110 --> 00:14:03,930 So the shape of table salt makes a 266 00:14:03,930 --> 00:14:06,260 difference in how it behaves. 267 00:14:06,260 --> 00:14:09,250 And the way you change the shape is to find something 268 00:14:09,250 --> 00:14:12,760 that will be absorbed preferentially on one set of 269 00:14:12,760 --> 00:14:17,060 faces that you would like to have be the faces that appear 270 00:14:17,060 --> 00:14:18,470 on the crystal. 271 00:14:18,470 --> 00:14:22,980 And that will do the job for you. 272 00:14:22,980 --> 00:14:27,070 I'll close with just one final example of this in real, every 273 00:14:27,070 --> 00:14:29,230 day industry or technology. 274 00:14:29,230 --> 00:14:35,790 I once had a phone call from a company that made salt for 275 00:14:35,790 --> 00:14:38,240 sprinkling on roads in the winter time, particularly 276 00:14:38,240 --> 00:14:41,830 around New England when the roads tend to ice up. 277 00:14:41,830 --> 00:14:46,070 And in an earlier era it, was politically correct to use 278 00:14:46,070 --> 00:14:47,760 sodium chloride. 279 00:14:47,760 --> 00:14:51,760 But you wanted the sodium chloride, if you could, to 280 00:14:51,760 --> 00:14:55,270 have an acicular, a needle-like, shape to improve 281 00:14:55,270 --> 00:14:57,680 its flow properties so you didn't waste a lot when you 282 00:14:57,680 --> 00:15:00,120 shook it out of the back of the truck. 283 00:15:00,120 --> 00:15:04,680 And the material that they use for that-- you think rock salt 284 00:15:04,680 --> 00:15:06,450 in the environment is bad-- 285 00:15:06,450 --> 00:15:11,580 they used a cyanide compound to modify the growth habit and 286 00:15:11,580 --> 00:15:13,430 make it needle-like. 287 00:15:13,430 --> 00:15:15,970 And they wanted to get away from that before they were 288 00:15:15,970 --> 00:15:22,330 really dragged into some sort of trial by the EPA. 289 00:15:22,330 --> 00:15:27,420 And did I know of something that could change the shape of 290 00:15:27,420 --> 00:15:29,025 the salt crystals for them? 291 00:15:29,025 --> 00:15:30,720 And the answer to that question was quite 292 00:15:30,720 --> 00:15:31,560 straightforward. 293 00:15:31,560 --> 00:15:32,450 No. 294 00:15:32,450 --> 00:15:34,100 That was the end of the conversation. 295 00:15:34,100 --> 00:15:36,760 But it was an interesting example of how surface 296 00:15:36,760 --> 00:15:39,090 chemistry can change crystal shape. 297 00:15:43,790 --> 00:15:44,260 OK. 298 00:15:44,260 --> 00:15:46,480 So all that long-winded explanation was a 299 00:15:46,480 --> 00:15:49,130 justification of the meaning of those 300 00:15:49,130 --> 00:15:51,490 two final two puzzles. 301 00:15:51,490 --> 00:15:54,010 And the final thing I have to say about them is that these 302 00:15:54,010 --> 00:15:57,810 puzzles were lifted from a puzzle book 303 00:15:57,810 --> 00:16:00,030 published by MENSA. 304 00:16:00,030 --> 00:16:02,790 Have any of you heard of MENSA? 305 00:16:02,790 --> 00:16:07,880 MENSA is an association of self-declared geniuses. 306 00:16:07,880 --> 00:16:10,120 So this was a puzzle book for geniuses. 307 00:16:10,120 --> 00:16:14,200 And you all did horribly well on that third problem set. 308 00:16:14,200 --> 00:16:16,810 Everybody at MIT is bright. 309 00:16:16,810 --> 00:16:20,410 But you folks are not only bright, you are geniuses 310 00:16:20,410 --> 00:16:24,294 because you cracked the MENSA puzzles. 311 00:16:24,294 --> 00:16:25,270 OK. 312 00:16:25,270 --> 00:16:27,080 Enough enjoyment and fun and games. 313 00:16:27,080 --> 00:16:31,070 Let's get serious, back down to what we were doing earlier. 314 00:16:31,070 --> 00:16:35,320 We had established in earlier discussion that there are a 315 00:16:35,320 --> 00:16:43,800 very limited number of ways in which we can arrange a mirror 316 00:16:43,800 --> 00:16:46,040 plane and a rotation axis about one 317 00:16:46,040 --> 00:16:47,690 fixed point in space. 318 00:16:47,690 --> 00:16:51,440 And these, accordingly, are called the point groups. 319 00:16:51,440 --> 00:16:54,640 And in particular, they are the two-dimensional 320 00:16:54,640 --> 00:16:56,215 crystallographic point groups. 321 00:17:03,100 --> 00:17:08,859 We had rotation axes by themselves, 1, 2, 3, 4, and 6. 322 00:17:08,859 --> 00:17:11,730 And these are the only ones we need worry about because these 323 00:17:11,730 --> 00:17:14,720 are the only rotational symmetries that are compatible 324 00:17:14,720 --> 00:17:16,369 with translation. 325 00:17:16,369 --> 00:17:17,679 Then we had a mirror plane. 326 00:17:20,790 --> 00:17:24,240 And then there was no reason why we could not combine a 327 00:17:24,240 --> 00:17:27,319 mirror plane with these rotational symmetries. 328 00:17:27,319 --> 00:17:31,500 And in doing so, we used a theorem that said if you take 329 00:17:31,500 --> 00:17:35,590 two reflection operations, sigma 1 and sigma 2-- and 330 00:17:35,590 --> 00:17:38,510 again, I use sigma to represent an individual 331 00:17:38,510 --> 00:17:41,350 operation of reflection-- 332 00:17:41,350 --> 00:17:45,540 and a mirror plane is the locus of operations of 333 00:17:45,540 --> 00:17:47,415 reflection, and then reflecting back again, there 334 00:17:47,415 --> 00:17:49,200 are two operations involved. 335 00:17:49,200 --> 00:17:53,920 And if these are combined at some angle mu, sigma 1 336 00:17:53,920 --> 00:17:58,480 followed by sigma 2, when they are separated by an angle mu 337 00:17:58,480 --> 00:18:02,940 is the same as the net transformation of a rotation 338 00:18:02,940 --> 00:18:06,300 about their point of intersection through twice the 339 00:18:06,300 --> 00:18:09,780 angle which the mirror planes are combined. 340 00:18:09,780 --> 00:18:11,560 And that led us-- 341 00:18:11,560 --> 00:18:14,730 in addition to a one-fold axis, a two-fold access, a 342 00:18:14,730 --> 00:18:18,260 three-fold axis, a four-fold axis, and a six-fold axis-- 343 00:18:18,260 --> 00:18:22,440 let us combine reflection with these axes to get symmetry 344 00:18:22,440 --> 00:18:26,080 combinations that we named according to the symmetry 345 00:18:26,080 --> 00:18:29,130 elements which were present in the final combination. 346 00:18:29,130 --> 00:18:34,990 We used two mirror planes if the mirror planes that arose 347 00:18:34,990 --> 00:18:38,410 where independent. 348 00:18:38,410 --> 00:18:41,770 Independent in the sense that the two-fold axis never turned 349 00:18:41,770 --> 00:18:42,750 one into another. 350 00:18:42,750 --> 00:18:45,750 And independent in the sense that if we draw a 351 00:18:45,750 --> 00:18:50,970 representative motif, the way those motifs were disposed 352 00:18:50,970 --> 00:18:53,900 about one mirror plane was different from the way they 353 00:18:53,900 --> 00:18:56,730 were arranged about the other one. 354 00:18:56,730 --> 00:19:00,830 For a four-fold axis, we could add mirror planes. 355 00:19:00,830 --> 00:19:03,830 And they would have to be at half the angle of the 356 00:19:03,830 --> 00:19:07,380 four-fold axis. 357 00:19:07,380 --> 00:19:09,450 And as we saw, it was kind of a mouthful 358 00:19:09,450 --> 00:19:10,700 to call this 4MMMMMMMM. 359 00:19:13,110 --> 00:19:16,210 There were just two kinds of mirror planes, so this one was 360 00:19:16,210 --> 00:19:18,240 called 4MM. 361 00:19:18,240 --> 00:19:23,300 6MM, analogously, was a six-fold axis with mirror 362 00:19:23,300 --> 00:19:26,490 planes separated by 30 degrees, half the throw of the 363 00:19:26,490 --> 00:19:28,400 six-fold axis. 364 00:19:28,400 --> 00:19:35,370 And the one that I left out is 3, not m m but 3M because a 365 00:19:35,370 --> 00:19:41,540 three-fold axis has mirror planes 366 00:19:41,540 --> 00:19:43,750 separated by 60 degrees. 367 00:19:43,750 --> 00:19:49,190 And the same thing goes on at each of these mirror planes, 368 00:19:49,190 --> 00:19:52,730 pair of objects hanging at one end of the mirror plane, the 369 00:19:52,730 --> 00:19:54,730 other end is unadorned. 370 00:19:54,730 --> 00:19:56,180 So they all behave the same. 371 00:19:56,180 --> 00:19:58,060 There's only one kind of mirror plane. 372 00:19:58,060 --> 00:20:01,530 The three-fold axis maps one mirror plane into another, and 373 00:20:01,530 --> 00:20:03,360 so they have to be the same. 374 00:20:03,360 --> 00:20:11,130 So therefore, only one M appears in the symbol. 375 00:20:11,130 --> 00:20:17,050 Independent of this, we had, when we combined reflection or 376 00:20:17,050 --> 00:20:21,230 rotation with a lattice, found that there were a very limited 377 00:20:21,230 --> 00:20:24,990 number of lattices in two dimensions, the oblique 378 00:20:24,990 --> 00:20:33,560 lattice which had T1 not equal to T2 and T1 inclined to T2 by 379 00:20:33,560 --> 00:20:36,690 some general angle. 380 00:20:36,690 --> 00:20:41,600 The next degree of specialization was for a 381 00:20:41,600 --> 00:20:44,580 two-fold axis-- so either a one-fold or a two-fold axis 382 00:20:44,580 --> 00:20:46,622 could fit in here. 383 00:20:46,622 --> 00:20:51,960 For a three-fold axis or a six-fold axis, we had to have 384 00:20:51,960 --> 00:20:56,390 a lattice in which the two translations were identical to 385 00:20:56,390 --> 00:21:01,230 one another, identical in magnitude and identical in the 386 00:21:01,230 --> 00:21:05,060 way the atoms were ranged relative to these 387 00:21:05,060 --> 00:21:06,300 translations. 388 00:21:06,300 --> 00:21:10,740 And the angle between T1 over T2 had to be 389 00:21:10,740 --> 00:21:13,560 identically 120 degrees. 390 00:21:13,560 --> 00:21:17,920 And by convention, we take the larger of two angles, 120 391 00:21:17,920 --> 00:21:21,806 rather than 60, to define the inter axial angle. 392 00:21:21,806 --> 00:21:27,130 A four-fold axis required a net that was exactly square, 393 00:21:27,130 --> 00:21:30,240 not approximately square but exactly square. 394 00:21:30,240 --> 00:21:33,930 So T1 and T2 had to be identical in magnitude. 395 00:21:33,930 --> 00:21:38,960 The angle between them had to be exactly 90 degrees. 396 00:21:38,960 --> 00:21:40,300 And close is no cigar. 397 00:21:40,300 --> 00:21:43,700 It has to be exactly 90 degrees because that angle is 398 00:21:43,700 --> 00:21:45,700 generated by symmetry. 399 00:21:45,700 --> 00:21:48,930 And then at the place in which I'm gonna pick up today in 400 00:21:48,930 --> 00:21:53,975 just a moment is the case of a mirror plane and reflection, 401 00:21:53,975 --> 00:21:57,970 we saw depending on how we arranged the translations 402 00:21:57,970 --> 00:22:02,090 relative to the reflection plane, could give us either a 403 00:22:02,090 --> 00:22:05,290 lattice in the shape of a rectangle-- 404 00:22:05,290 --> 00:22:11,480 and here T1 and T2 made an angle of exactly 90 degrees. 405 00:22:11,480 --> 00:22:15,210 But the magnitude of T1 could be anything it liked relative 406 00:22:15,210 --> 00:22:20,400 to the magnitude of T2, and then a double cell, which had 407 00:22:20,400 --> 00:22:24,140 the shape of a diamond. 408 00:22:24,140 --> 00:22:28,820 But it was operationally much to our advantage to pick a 409 00:22:28,820 --> 00:22:33,200 double cell which had a right angle in it. 410 00:22:33,200 --> 00:22:36,940 And the reason for that was that it is a nightmare to do 411 00:22:36,940 --> 00:22:40,750 calculations of inter-atomic distances and angles or angles 412 00:22:40,750 --> 00:22:44,040 between faces in an oblique system. 413 00:22:44,040 --> 00:22:47,980 The right angle makes these calculations very simple. 414 00:22:47,980 --> 00:22:51,270 And the fact that this cell tells you twice as much of the 415 00:22:51,270 --> 00:22:55,330 area as you really need to know about is a 416 00:22:55,330 --> 00:22:56,650 small price to pay. 417 00:22:56,650 --> 00:22:59,810 So this is the rectangular net, as 418 00:22:59,810 --> 00:23:01,550 we'll call it in words. 419 00:23:04,830 --> 00:23:06,480 And this is one that we'll call the 420 00:23:06,480 --> 00:23:07,730 centered rectangular net. 421 00:23:14,690 --> 00:23:15,170 OK. 422 00:23:15,170 --> 00:23:17,660 So let me pause here and suck in air and see if there are 423 00:23:17,660 --> 00:23:19,766 any questions. 424 00:23:19,766 --> 00:23:21,590 This is where we left off. 425 00:23:21,590 --> 00:23:24,500 And now having everything spread out on the board, we're 426 00:23:24,500 --> 00:23:26,320 going to start to make combinations and 427 00:23:26,320 --> 00:23:27,310 continue that process. 428 00:23:27,310 --> 00:23:28,090 Yes, sir. 429 00:23:28,090 --> 00:23:29,743 AUDIENCE: So the centered rectangular has the same exact 430 00:23:29,743 --> 00:23:31,961 [? intercept ?] as the rectangular but the only thing 431 00:23:31,961 --> 00:23:34,180 you're doing is you're deriving it based on symmetry 432 00:23:34,180 --> 00:23:36,645 that you're using like a diamond kind of thing to get 433 00:23:36,645 --> 00:23:37,138 the center one? 434 00:23:37,138 --> 00:23:37,640 PROFESSOR: Yeah. 435 00:23:37,640 --> 00:23:41,760 So all these arose when we did it slowly and systematically. 436 00:23:41,760 --> 00:23:45,930 We said let's let this be the location of the mirror plane. 437 00:23:45,930 --> 00:23:49,210 What happens if we combine it with a translation? 438 00:23:49,210 --> 00:23:51,820 We can take one lattice point on the mirror plane since 439 00:23:51,820 --> 00:23:53,740 there's no unique lattice point. 440 00:23:53,740 --> 00:23:55,530 And then the mirror plane is gonna reflect 441 00:23:55,530 --> 00:23:57,710 this over to here. 442 00:23:57,710 --> 00:23:59,340 So here's a first translation. 443 00:23:59,340 --> 00:24:02,170 I've got a second non-colinear translation. 444 00:24:02,170 --> 00:24:03,420 Wham-o. 445 00:24:03,420 --> 00:24:04,530 Instant lattice. 446 00:24:04,530 --> 00:24:09,720 So what I'm doing is taking this diamond shape and I am 447 00:24:09,720 --> 00:24:13,530 taking one translation here-- let's put some labels on this. 448 00:24:13,530 --> 00:24:16,870 Let's call this T1 and this T2. 449 00:24:16,870 --> 00:24:21,930 So this translation here, call it T1 prime, is my original 450 00:24:21,930 --> 00:24:24,490 translation T1 plus T2. 451 00:24:24,490 --> 00:24:30,778 In this translation, T2 is the negative of the-- 452 00:24:30,778 --> 00:24:31,004 Yeah. 453 00:24:31,004 --> 00:24:34,660 It's gonna be minus T1 plus T2. 454 00:24:34,660 --> 00:24:37,130 So I've taken linear combinations of the two 455 00:24:37,130 --> 00:24:38,950 vectors of the diamond-shaped cell. 456 00:24:41,590 --> 00:24:46,870 And again, this is contrary to the rules we have that say 457 00:24:46,870 --> 00:24:48,980 pick the shortest two translations. 458 00:24:48,980 --> 00:24:50,010 Why? 459 00:24:50,010 --> 00:24:52,295 The final larger area of volume then you need. 460 00:24:52,295 --> 00:24:56,990 And the answer to that is occasionally the provision of 461 00:24:56,990 --> 00:25:00,970 a coordinate system, which is far easier to work in, is a 462 00:25:00,970 --> 00:25:04,375 small price to pay for that redundancy. 463 00:25:04,375 --> 00:25:04,800 OK. 464 00:25:04,800 --> 00:25:06,770 And then the primitive rectangular net, just to 465 00:25:06,770 --> 00:25:09,855 remind you again, we said is there any case in which this 466 00:25:09,855 --> 00:25:11,100 is not true? 467 00:25:11,100 --> 00:25:15,380 And that case was if you take T1 exactly perpendicular to 468 00:25:15,380 --> 00:25:19,200 the mirror plane and then you define only a lattice row. 469 00:25:19,200 --> 00:25:22,110 So that gives you a second choice for T2. 470 00:25:22,110 --> 00:25:24,960 You can't pick it anywhere you like, otherwise it gets 471 00:25:24,960 --> 00:25:27,970 reflected across just as in the first case, and you have 472 00:25:27,970 --> 00:25:31,800 two translations that are not compatible. 473 00:25:31,800 --> 00:25:35,850 And the only way that they are compatible is that if you make 474 00:25:35,850 --> 00:25:38,870 the translation T1 straddle the mirror plane. 475 00:25:38,870 --> 00:25:40,810 And that's what we've already got up here. 476 00:25:40,810 --> 00:25:42,870 So that's how we got the rectangular net. 477 00:25:42,870 --> 00:25:46,320 We could get a second distinct sort of lattice only if this 478 00:25:46,320 --> 00:25:49,460 translation was exactly in the plane of the mirror line. 479 00:25:49,460 --> 00:25:49,880 Yes, sir. 480 00:25:49,880 --> 00:25:53,039 AUDIENCE: But the number and kind of symmetry elements are 481 00:25:53,039 --> 00:25:54,740 the same for both, right? 482 00:25:54,740 --> 00:25:56,684 PROFESSOR: Exactly. 483 00:25:56,684 --> 00:25:58,640 AUDIENCE: Uh, so-- 484 00:25:58,640 --> 00:26:01,670 PROFESSOR: So there's a curious sort of duality here. 485 00:26:01,670 --> 00:26:06,610 Here in the case of a hexagonal net, this is just a 486 00:26:06,610 --> 00:26:10,030 lattice of a very specialized shape. 487 00:26:10,030 --> 00:26:12,790 And does that have a six-fold axis in it? 488 00:26:12,790 --> 00:26:14,870 No, not unless I decide to put one in. 489 00:26:14,870 --> 00:26:17,660 I could put in a three-fold axis, alternatively. 490 00:26:17,660 --> 00:26:20,700 So here, I have one lattice that can accept two different 491 00:26:20,700 --> 00:26:21,900 kinds of symmetry. 492 00:26:21,900 --> 00:26:23,950 Here I've got the reverse situation. 493 00:26:23,950 --> 00:26:27,500 I have two distinct kind of lattices, both of which are 494 00:26:27,500 --> 00:26:31,860 happy and content with the same symmetry on them. 495 00:26:31,860 --> 00:26:35,560 So there's going to be a far greater number of combinations 496 00:26:35,560 --> 00:26:39,180 of lattice with symmetry than simply a one-to-one 497 00:26:39,180 --> 00:26:43,600 correspondence between lattice types and point group types. 498 00:26:49,040 --> 00:26:49,440 OK. 499 00:26:49,440 --> 00:26:50,690 Any other questions? 500 00:26:53,120 --> 00:26:58,460 All right so let me now shift down to low gear and remind 501 00:26:58,460 --> 00:27:03,930 you of one thing that we did last time. 502 00:27:03,930 --> 00:27:14,060 We asked what happens if I can combine a rotation operation, 503 00:27:14,060 --> 00:27:18,690 a alpha, with a translation? 504 00:27:18,690 --> 00:27:21,510 Call this T1. 505 00:27:21,510 --> 00:27:25,900 Sounds like something I already did in showing that 506 00:27:25,900 --> 00:27:30,490 the values of alpha are restricted to values. 507 00:27:30,490 --> 00:27:34,480 But what I'm going to do now is use something that looks as 508 00:27:34,480 --> 00:27:38,220 though it's a similar starting point to arrive at a different 509 00:27:38,220 --> 00:27:41,270 result that we saw last time. 510 00:27:41,270 --> 00:27:47,760 I'm going to deliberately look at a line that is alpha over 2 511 00:27:47,760 --> 00:27:52,410 on one side of the perpendicular to T1. 512 00:27:52,410 --> 00:27:58,730 And then the operation of the a alpha is going to take this 513 00:27:58,730 --> 00:28:01,930 translation with a lattice point of necessity at the end 514 00:28:01,930 --> 00:28:06,530 of it, and it's going to move it over to here, alpha over 2, 515 00:28:06,530 --> 00:28:10,540 on the other side of the translation of the 516 00:28:10,540 --> 00:28:12,470 perpendicular. 517 00:28:12,470 --> 00:28:15,850 Then I'm going to pick this up with the translation T1 and 518 00:28:15,850 --> 00:28:21,050 move it and everything along it to a location like this. 519 00:28:21,050 --> 00:28:28,280 And my question now is, what is a alpha followed by the 520 00:28:28,280 --> 00:28:31,805 translation T1? 521 00:28:31,805 --> 00:28:34,540 The handedness of the objects have not changed. 522 00:28:34,540 --> 00:28:35,900 So this has to be either 523 00:28:35,900 --> 00:28:39,530 translation or another rotation. 524 00:28:39,530 --> 00:28:43,510 And what we can very quickly show is if upon dropping a 525 00:28:43,510 --> 00:28:48,050 perpendicular down to the original translation, these 526 00:28:48,050 --> 00:28:49,310 lines are parallel. 527 00:28:49,310 --> 00:28:50,800 This line cuts across it. 528 00:28:50,800 --> 00:28:54,470 So this angle is alpha over 2. 529 00:28:54,470 --> 00:28:58,730 And this line is inclined to the perpendicular 530 00:28:58,730 --> 00:29:00,140 by alpha over 2. 531 00:29:00,140 --> 00:29:05,580 So this angle in here is also alpha over 2. 532 00:29:05,580 --> 00:29:11,900 So the answer is that if I rotate from one location to 533 00:29:11,900 --> 00:29:16,420 another by rotation alpha and then pick up that motif or the 534 00:29:16,420 --> 00:29:22,710 entire space and translate it by the original translation T1 535 00:29:22,710 --> 00:29:26,010 to get a third one-- this is number 1, right-handed, say, 536 00:29:26,010 --> 00:29:28,040 number 2, right-handed. 537 00:29:28,040 --> 00:29:29,150 This is number 3. 538 00:29:29,150 --> 00:29:30,470 Stays right-handed. 539 00:29:30,470 --> 00:29:32,750 Has to be related by a rotation. 540 00:29:32,750 --> 00:29:37,540 And we next ask, what point is the point about which the 541 00:29:37,540 --> 00:29:39,230 rotation occurs? 542 00:29:39,230 --> 00:29:40,110 Then we used-- 543 00:29:40,110 --> 00:29:44,240 what at the time seemed like a trivial observation-- 544 00:29:44,240 --> 00:29:47,830 that a rotation axis in space or a rotation point in two 545 00:29:47,830 --> 00:29:51,080 dimensions is the locus that is left 546 00:29:51,080 --> 00:29:55,500 unmoved by the rotation. 547 00:29:55,500 --> 00:30:00,460 So if rotating from here and translating over to here is 548 00:30:00,460 --> 00:30:03,770 equivalent to a rotation and if the locus of the rotation 549 00:30:03,770 --> 00:30:08,530 has to be the point that's left on moved, bingo. 550 00:30:08,530 --> 00:30:11,370 That's where the rotation occurs. 551 00:30:11,370 --> 00:30:14,260 And it's through the same angle, so we can label this an 552 00:30:14,260 --> 00:30:16,870 operation b alpha. 553 00:30:16,870 --> 00:30:21,050 And we know exactly where that point is gonna be. 554 00:30:21,050 --> 00:30:24,080 It's gonna be along the perpendicular bisector of the 555 00:30:24,080 --> 00:30:25,300 original translation. 556 00:30:25,300 --> 00:30:30,180 And it's gonna be up a distance x which is T over 2 557 00:30:30,180 --> 00:30:33,185 times the cotangent of alpha over 2. 558 00:30:33,185 --> 00:30:38,490 So now we've got another, what I call, combination theorems. 559 00:30:38,490 --> 00:30:42,260 And again, this is nothing more than knowing how to 560 00:30:42,260 --> 00:30:46,490 complete the product of two operations in establishing the 561 00:30:46,490 --> 00:30:48,130 group multiplication table. 562 00:30:48,130 --> 00:30:51,970 so let me write it down in the form of a combination theorem. 563 00:30:51,970 --> 00:30:59,060 This says that a alpha followed by a translation is 564 00:30:59,060 --> 00:31:01,570 another rotation-- 565 00:31:01,570 --> 00:31:04,190 in the same sense, both counter 566 00:31:04,190 --> 00:31:06,550 clockwise in this example-- 567 00:31:06,550 --> 00:31:10,280 another rotation b alpha in the same sense about a point 568 00:31:10,280 --> 00:31:15,550 that is always T over 2 times the cotangent of alpha over 2 569 00:31:15,550 --> 00:31:18,310 along the perpendicular bisector. 570 00:31:18,310 --> 00:31:20,870 So that's how you would go about making an entry in the 571 00:31:20,870 --> 00:31:24,050 group multiplication table. 572 00:31:24,050 --> 00:31:24,260 Sorry. 573 00:31:24,260 --> 00:31:24,890 I put you off. 574 00:31:24,890 --> 00:31:25,316 AUDIENCE: Yeah. 575 00:31:25,316 --> 00:31:27,840 Can you just, like, I just want to see what you did 576 00:31:27,840 --> 00:31:30,830 actually to the motif in that drawing. 577 00:31:30,830 --> 00:31:31,310 PROFESSOR: OK. 578 00:31:31,310 --> 00:31:35,520 Rather than taking an abstraction, an arbitrary line 579 00:31:35,520 --> 00:31:40,210 that is at alpha over 2, I put a motif in the space that 580 00:31:40,210 --> 00:31:44,240 doesn't necessarily hang on this particular locus that I 581 00:31:44,240 --> 00:31:45,550 drew for reference. 582 00:31:45,550 --> 00:31:49,650 And the rotation through alpha would move it to here. 583 00:31:49,650 --> 00:31:51,600 It's lagging behind this first line. 584 00:31:51,600 --> 00:31:54,550 It lagged behind the second one by the same amount. 585 00:31:54,550 --> 00:31:57,850 Then I pick it up and I move it by the same translation and 586 00:31:57,850 --> 00:32:02,780 that slides it over here, still canted over to the left. 587 00:32:02,780 --> 00:32:06,260 And now what I claim is that I get from the first one to the 588 00:32:06,260 --> 00:32:07,170 third one-- 589 00:32:07,170 --> 00:32:09,740 and actually should have raised your hand on a 590 00:32:09,740 --> 00:32:10,470 different matter. 591 00:32:10,470 --> 00:32:12,540 This is out in front of the translation. 592 00:32:12,540 --> 00:32:15,380 So number 3 should sit here, also to the right to the 593 00:32:15,380 --> 00:32:16,720 translation. 594 00:32:16,720 --> 00:32:19,590 And the way I get from 1 to 3 in one shot is by 595 00:32:19,590 --> 00:32:20,919 rotating alpha b. 596 00:32:23,913 --> 00:32:26,408 OK. 597 00:32:26,408 --> 00:32:31,840 Now, a cautionary note just so you do not use your new power 598 00:32:31,840 --> 00:32:33,330 recklessly. 599 00:32:33,330 --> 00:32:47,710 Let me emphasize that this is an escalation in operations, 600 00:32:47,710 --> 00:32:49,183 not in symmetry elements. 601 00:32:59,260 --> 00:33:04,990 And the reason for that is that a rotation axis, in 602 00:33:04,990 --> 00:33:08,190 general, contains a number of different rotations. 603 00:33:08,190 --> 00:33:12,840 An n-fold axis has n different rotation operations implied in 604 00:33:12,840 --> 00:33:15,760 it, including the identity operation of 605 00:33:15,760 --> 00:33:17,850 rotating 360 degrees. 606 00:33:17,850 --> 00:33:21,460 So if I say I'm gonna drop a four-fold access in a lattice, 607 00:33:21,460 --> 00:33:24,570 what I'm doing is dropping into the lattice a 90 degree 608 00:33:24,570 --> 00:33:27,430 rotation, a 180 degree rotation, and 609 00:33:27,430 --> 00:33:30,180 a 270 degree rotation. 610 00:33:30,180 --> 00:33:35,500 And the location of the new operations depends on alpha. 611 00:33:35,500 --> 00:33:38,050 So these new operations are not gonna pop 612 00:33:38,050 --> 00:33:39,560 up at the same location. 613 00:33:39,560 --> 00:33:43,660 They're gonna be sprinkled over different locations. 614 00:33:43,660 --> 00:33:46,740 So as an abstraction, that may be hard to appreciate. 615 00:33:46,740 --> 00:33:49,960 But we're going to straight away derive a 616 00:33:49,960 --> 00:33:52,560 couple of more additions. 617 00:33:52,560 --> 00:33:56,320 And then when we see what happens upon adding a rotation 618 00:33:56,320 --> 00:34:00,570 axis to a lattice in a couple of nontrivial cases, I'll just 619 00:34:00,570 --> 00:34:03,710 summarize the results and assume that you'll be able to 620 00:34:03,710 --> 00:34:06,450 derive them on your own. 621 00:34:06,450 --> 00:34:09,120 The last time we had got around to doing just one of 622 00:34:09,120 --> 00:34:11,460 these combinations. 623 00:34:11,460 --> 00:34:16,730 And we ask what happens when you combine with a translation 624 00:34:16,730 --> 00:34:20,290 a rotation operation a pi. 625 00:34:20,290 --> 00:34:24,760 So we'll take some motif that sets up here, right-handed, 626 00:34:24,760 --> 00:34:29,960 rotate it by 180 degrees to get a second one, and then 627 00:34:29,960 --> 00:34:35,020 pick that up and translate it by T. So I'll get a third one 628 00:34:35,020 --> 00:34:39,650 sitting down like this of the same handedness. 629 00:34:39,650 --> 00:34:41,750 And the question now is how do I get from 630 00:34:41,750 --> 00:34:43,449 1 to number 3 directly? 631 00:34:43,449 --> 00:34:45,020 Well, let's use our theorem. 632 00:34:45,020 --> 00:34:48,010 Our theorem says we should go a distance x along the 633 00:34:48,010 --> 00:34:52,989 perpendicular bisector, which is half the magnitude of T 634 00:34:52,989 --> 00:34:57,310 times the cotangent of pi over 2. 635 00:34:57,310 --> 00:35:00,120 The cotangent of pi over 2 is zero. 636 00:35:00,120 --> 00:35:01,760 So we go up a distance x. 637 00:35:01,760 --> 00:35:04,860 That's equal to 0, all in the perpendicular bisector, which 638 00:35:04,860 --> 00:35:06,680 means we stay put. 639 00:35:06,680 --> 00:35:08,860 And-- 640 00:35:08,860 --> 00:35:13,060 in this class, there is always truth in advertising. 641 00:35:13,060 --> 00:35:18,580 Around this locus here is a rotation operation b pi, which 642 00:35:18,580 --> 00:35:22,820 takes the first one into the third one. 643 00:35:22,820 --> 00:35:25,720 OK. 644 00:35:25,720 --> 00:35:30,450 And now, lickety split, I will derive again-- and I apologize 645 00:35:30,450 --> 00:35:33,500 for going fast because I did do it last time. 646 00:35:33,500 --> 00:35:37,730 Let's ask what happens when we combine a two-fold axis with a 647 00:35:37,730 --> 00:35:40,160 parallelogram net? 648 00:35:40,160 --> 00:35:44,030 So we're taking a two-fold axis plus a parallelogram net. 649 00:35:47,020 --> 00:35:49,070 In this two dimensional lattice there are two 650 00:35:49,070 --> 00:35:54,880 translations, T1 and T2, of arbitrary magnitude with some 651 00:35:54,880 --> 00:35:56,960 angle between them that's arbitrary. 652 00:35:56,960 --> 00:36:01,440 And to this lattice point, I'll add a two-fold axis. 653 00:36:01,440 --> 00:36:06,880 A two-fold axis involves the presence of two operations, 654 00:36:06,880 --> 00:36:10,490 the identity operation, and that we can forget about, and 655 00:36:10,490 --> 00:36:14,520 then there's the operation a pi. 656 00:36:14,520 --> 00:36:19,250 By combine a pi with T1, I'll get an operation b pi that 657 00:36:19,250 --> 00:36:20,910 sits at the midpoint of T1. 658 00:36:20,910 --> 00:36:28,340 If I combine a pi with T2, we get an operation c pi at the 659 00:36:28,340 --> 00:36:29,900 midpoint of T2. 660 00:36:29,900 --> 00:36:37,540 If I combine a pi with T1 plus T2, I'll get another operation 661 00:36:37,540 --> 00:36:40,680 d pi at the center of the cell. 662 00:36:40,680 --> 00:36:44,540 And we don't have to ask what happens when we go three 663 00:36:44,540 --> 00:36:46,160 translations over. 664 00:36:46,160 --> 00:36:49,730 There's gonna be another operation of 180 degree 665 00:36:49,730 --> 00:36:51,340 rotation on that translation. 666 00:36:51,340 --> 00:36:55,420 But we really only need bother about what goes on on the 667 00:36:55,420 --> 00:36:58,720 edges and in the interior of one unit cell, because 668 00:36:58,720 --> 00:37:02,340 whatever's there has to be repeated by translation. 669 00:37:02,340 --> 00:37:15,270 So, in making these additions, we need consider only 670 00:37:15,270 --> 00:37:16,740 independent translations-- 671 00:37:16,740 --> 00:37:19,110 that is, not related to one another by 672 00:37:19,110 --> 00:37:20,660 the rotational symmetry-- 673 00:37:20,660 --> 00:37:28,050 independent translations that terminate 674 00:37:28,050 --> 00:37:29,300 within the unit cell. 675 00:37:40,910 --> 00:37:45,860 And that means for the addition of a two-fold axis, I 676 00:37:45,860 --> 00:37:48,570 need do only what I have, in fact, just done. 677 00:37:48,570 --> 00:37:57,850 I want to consider a pi with T1, T2, and T1 plus T2. 678 00:38:02,180 --> 00:38:07,010 If I have the operation a pi or b pi or c pi or d pi at 679 00:38:07,010 --> 00:38:15,720 these four locations, that is all I need have present to say 680 00:38:15,720 --> 00:38:19,250 that I have, in addition to the two-fold axis that I added 681 00:38:19,250 --> 00:38:23,390 to the lattice point, a two-fold axis halfway along 682 00:38:23,390 --> 00:38:28,230 T1, a two-fold access halfway along T2, and another one 683 00:38:28,230 --> 00:38:30,680 smack in the middle of the cell. 684 00:38:30,680 --> 00:38:32,140 These are lattice points. 685 00:38:32,140 --> 00:38:36,050 So those axes have to be repeated by translations, this 686 00:38:36,050 --> 00:38:40,560 one from here to here, this one from here, this one from 687 00:38:40,560 --> 00:38:41,600 here to here. 688 00:38:41,600 --> 00:38:45,240 So I'll have two-fold axes in the middle of all of the edges 689 00:38:45,240 --> 00:38:51,150 of the cell plus this fourth one right in the middle. 690 00:38:51,150 --> 00:38:56,610 So this is an example of a two dimensional space group. 691 00:38:56,610 --> 00:39:01,470 It's a symmetry that acts on all of space. 692 00:39:01,470 --> 00:39:10,400 And the names that we will give to these combinations is 693 00:39:10,400 --> 00:39:14,820 a symbol for the symmetry element that we added, in this 694 00:39:14,820 --> 00:39:16,930 case a two-fold axis. 695 00:39:16,930 --> 00:39:21,480 And then we'll specify the nature of the lattice to which 696 00:39:21,480 --> 00:39:22,630 we've added it. 697 00:39:22,630 --> 00:39:25,930 And all that I have to say is that the lattice is primitive 698 00:39:25,930 --> 00:39:30,200 because you're all aware at this point that a two-fold 699 00:39:30,200 --> 00:39:33,540 axis requires only a primitive, a 700 00:39:33,540 --> 00:39:34,990 parallelogram net. 701 00:39:34,990 --> 00:39:37,770 The only place we'll need a special symbol is when we add 702 00:39:37,770 --> 00:39:40,610 a mirror plane to either the rectangular net or the 703 00:39:40,610 --> 00:39:41,910 centered rectangular net. 704 00:39:41,910 --> 00:39:45,450 And then we'll have to specify which type of lattice of that 705 00:39:45,450 --> 00:39:48,023 shape, primitive or centered, we're dealing with. 706 00:39:48,023 --> 00:39:48,790 Yes, sir. 707 00:39:48,790 --> 00:39:50,511 AUDIENCE: So all you did with that derivation there is prove 708 00:39:50,511 --> 00:39:52,115 that you don't have to worry about any 709 00:39:52,115 --> 00:39:54,039 rotations outside of that? 710 00:39:54,039 --> 00:39:54,520 PROFESSOR: That's correct. 711 00:39:54,520 --> 00:39:55,482 AUDIENCE: That's correct? 712 00:39:55,482 --> 00:39:57,410 PROFESSOR: Yep. 713 00:39:57,410 --> 00:39:59,640 Which is very fortunate 'cause there's an infinite number 714 00:39:59,640 --> 00:40:02,730 lurking outside the boundaries of the cell. 715 00:40:02,730 --> 00:40:05,300 I'll show you another general truth. 716 00:40:05,300 --> 00:40:07,650 What does a pattern look like? 717 00:40:07,650 --> 00:40:12,780 The pattern looks exactly like the pattern of a two-fold axis 718 00:40:12,780 --> 00:40:16,530 with that pattern of motifs hung at every lattice point. 719 00:40:23,780 --> 00:40:29,250 These new two-fold axes don't do any further repetition of 720 00:40:29,250 --> 00:40:31,610 the motifs. 721 00:40:31,610 --> 00:40:34,040 They don't do any repetition of the motif. 722 00:40:34,040 --> 00:40:38,770 They just express relation between things that you get 723 00:40:38,770 --> 00:40:41,820 when you take this pair and then repeat it by the 724 00:40:41,820 --> 00:40:43,250 translations. 725 00:40:43,250 --> 00:40:46,500 So this two-fold access, for example, relates this to this. 726 00:40:46,500 --> 00:40:49,460 This two-fold axis relates this one to this one, and this 727 00:40:49,460 --> 00:40:51,130 one to this one, and so on. 728 00:40:51,130 --> 00:40:54,280 They're just relations that exist between the things that 729 00:40:54,280 --> 00:40:57,260 you get when you add the pattern of a two-fold axis to 730 00:40:57,260 --> 00:40:59,090 a lattice point. 731 00:40:59,090 --> 00:41:02,420 So another generalization of what we're doing here that 732 00:41:02,420 --> 00:41:17,100 turns out to be valid, the pattern of motifs for a 733 00:41:17,100 --> 00:41:43,460 particular plane group is nothing more than the pattern 734 00:41:43,460 --> 00:41:44,760 of the symmetry element-- 735 00:41:51,540 --> 00:41:55,670 symmetry element or symmetry elements, sometimes we'll be 736 00:41:55,670 --> 00:41:56,920 adding more than one-- 737 00:42:08,890 --> 00:42:12,330 that we've added to a lattice point. 738 00:42:21,090 --> 00:42:25,330 So in other words, if we take 2MM and drop that into a 739 00:42:25,330 --> 00:42:27,700 rectangular lattice, the pattern is simply gonna be 740 00:42:27,700 --> 00:42:33,400 these four atoms related by 2MM hung at every corner of 741 00:42:33,400 --> 00:42:34,385 the lattice. 742 00:42:34,385 --> 00:42:37,700 If you add it to a centered lattice, these would be hung 743 00:42:37,700 --> 00:42:39,890 at every corner of the lattice, and also at the 744 00:42:39,890 --> 00:42:41,330 centered lattice point. 745 00:42:41,330 --> 00:42:42,790 So it's that simple. 746 00:42:42,790 --> 00:42:45,490 If you look at some of these high symmetry space groups or 747 00:42:45,490 --> 00:42:48,050 plane groups, wow, there's symmetry all over the place. 748 00:42:48,050 --> 00:42:49,960 You say, how can I draw a pattern for that? 749 00:42:49,960 --> 00:42:50,550 Easy. 750 00:42:50,550 --> 00:42:53,060 Just do what the point group does and drop it in at the 751 00:42:53,060 --> 00:42:55,300 lattice points. 752 00:42:55,300 --> 00:42:59,530 So that turns out to be a universal feature of all space 753 00:42:59,530 --> 00:43:00,780 groups and plane groups. 754 00:43:03,187 --> 00:43:03,670 OK. 755 00:43:03,670 --> 00:43:05,320 Somebody had a question over here and I cut you off. 756 00:43:05,320 --> 00:43:05,750 Yeah. 757 00:43:05,750 --> 00:43:06,180 AUDIENCE: Yeah. 758 00:43:06,180 --> 00:43:08,871 On that two [INAUDIBLE] diagram, wouldn't you want to 759 00:43:08,871 --> 00:43:11,642 use four of those because you said it was-- you don't 760 00:43:11,642 --> 00:43:14,255 concern ones that are-- you're only concerned independent 761 00:43:14,255 --> 00:43:15,690 ones, the ones you can't get by-- 762 00:43:15,690 --> 00:43:17,090 PROFESSOR: You're absolutely right. 763 00:43:17,090 --> 00:43:19,070 And let's look at this pattern again. 764 00:43:19,070 --> 00:43:21,610 Here are a pair that are hanging close to 765 00:43:21,610 --> 00:43:23,680 this two-fold axis. 766 00:43:23,680 --> 00:43:26,910 Here's a pair hanging close to the two-fold axis, very 767 00:43:26,910 --> 00:43:30,520 different arrangement, different arrangement relative 768 00:43:30,520 --> 00:43:32,780 to this two-fold axis, different arrangements 769 00:43:32,780 --> 00:43:34,260 relative to this two-fold axis. 770 00:43:34,260 --> 00:43:36,660 So what we're saying is-- and this is the way, in fact, we 771 00:43:36,660 --> 00:43:37,680 derive them-- 772 00:43:37,680 --> 00:43:41,640 that these four two-fold axes are all distinct. 773 00:43:41,640 --> 00:43:46,210 But the ones that are along the edges are just related by 774 00:43:46,210 --> 00:43:47,800 translation. 775 00:43:47,800 --> 00:43:49,420 So why do I have to put them in? 776 00:43:49,420 --> 00:43:52,240 AUDIENCE: You just put them in to just show the symmetry? 777 00:43:52,240 --> 00:43:52,710 PROFESSOR: Yeah. 778 00:43:52,710 --> 00:43:53,490 Exactly. 779 00:43:53,490 --> 00:43:57,080 Exactly. 'Cause if I say this is a lattice and I have 780 00:43:57,080 --> 00:44:03,900 two-fold axes here, that's an incomplete representation 781 00:44:03,900 --> 00:44:05,980 because if these are translations, I jolly well 782 00:44:05,980 --> 00:44:09,290 better have the same thing at all the lattice points. 783 00:44:09,290 --> 00:44:12,230 I'm just emphasizing the property of a lattice which is 784 00:44:12,230 --> 00:44:16,930 understood to be present in all of these combinations. 785 00:44:16,930 --> 00:44:19,330 OK. 786 00:44:19,330 --> 00:44:20,580 Other questions? 787 00:44:26,380 --> 00:44:30,060 And I would remind you that a reasonable question or request 788 00:44:30,060 --> 00:44:31,890 would be, could you do that all again, 789 00:44:31,890 --> 00:44:33,800 please, at half speed? 790 00:44:33,800 --> 00:44:35,700 And I'd be happy to comply. 791 00:44:35,700 --> 00:44:38,500 But really this is ground that we covered last time. 792 00:44:38,500 --> 00:44:40,100 And I'm just reinforcing it. 793 00:44:40,100 --> 00:44:42,650 So now that we've run out of time, we can go on to 794 00:44:42,650 --> 00:44:43,910 something new. 795 00:44:43,910 --> 00:44:46,940 Let me do, in the couple minutes remaining, let me make 796 00:44:46,940 --> 00:44:52,130 one more addition just to show you how things change when you 797 00:44:52,130 --> 00:44:54,930 add a rotation axis that has more than 798 00:44:54,930 --> 00:44:58,745 one rotation operation. 799 00:44:58,745 --> 00:45:01,970 Let me ask what happens when you add a 800 00:45:01,970 --> 00:45:05,430 four-fold axis to a lattice. 801 00:45:05,430 --> 00:45:10,050 We've seen that a four-fold axis can coexist with no 802 00:45:10,050 --> 00:45:15,940 lattice other than 1, which is dimensionally square and which 803 00:45:15,940 --> 00:45:22,940 has the same thing hanging about, two translations that 804 00:45:22,940 --> 00:45:27,400 are exactly 90 degrees apart. 805 00:45:27,400 --> 00:45:32,030 So what we're doing is taking a four-fold axis and drop it 806 00:45:32,030 --> 00:45:36,030 in at the corner lattice point and then step back before all 807 00:45:36,030 --> 00:45:37,872 hell breaks looks. 808 00:45:37,872 --> 00:45:40,510 Now, first thing we're gonna do is to get the four-fold 809 00:45:40,510 --> 00:45:44,980 axis at every corner lattice point. 810 00:45:44,980 --> 00:45:48,970 And then we're gonna use our theorem, taking into account 811 00:45:48,970 --> 00:45:52,910 that when I say I'm adding a four-fold axis, I am adding 812 00:45:52,910 --> 00:46:01,010 the operations a pi over 2, a pi, a3 pi over 2. 813 00:46:01,010 --> 00:46:06,490 But let me call that, instead, a minus pi over 2, the same 814 00:46:06,490 --> 00:46:08,900 thing and a little easier to deal with. 815 00:46:08,900 --> 00:46:11,860 And then the operation a2 pi, which is the same as the 816 00:46:11,860 --> 00:46:13,020 identity operation. 817 00:46:13,020 --> 00:46:14,950 And that's dull and uninteresting, so we won't 818 00:46:14,950 --> 00:46:17,580 consider that addition. 819 00:46:17,580 --> 00:46:22,820 We've already done this, a pi, in deriving P2. 820 00:46:22,820 --> 00:46:24,810 And that's the nice thing about this derivation. 821 00:46:24,810 --> 00:46:28,400 It's gonna snowball because as the symmetry gets higher, 822 00:46:28,400 --> 00:46:32,220 we've already done the work for the subgroup that contains 823 00:46:32,220 --> 00:46:34,955 the elements that are present in the higher symmetry. 824 00:46:34,955 --> 00:46:39,550 So we'll get an operation a pi over a pi here. 825 00:46:39,550 --> 00:46:42,010 We'll get an operation a pi here. 826 00:46:42,010 --> 00:46:45,950 And if we take the diagonal translation, T1 plus T2, I'll 827 00:46:45,950 --> 00:46:47,600 get the operation a pi here. 828 00:46:47,600 --> 00:46:52,260 Or I should probably call them a pi, b pi, and c pi to 829 00:46:52,260 --> 00:46:58,050 indicate that these are different two-fold axes. 830 00:46:58,050 --> 00:46:58,510 OK. 831 00:46:58,510 --> 00:47:03,580 Let me emphasize again that this theorem that we have is a 832 00:47:03,580 --> 00:47:06,330 theorem in individual operations and 833 00:47:06,330 --> 00:47:07,760 not symmetry elements. 834 00:47:07,760 --> 00:47:10,670 If I say a four-fold axis is present, it means that all of 835 00:47:10,670 --> 00:47:13,560 these operations exist about that locus. 836 00:47:13,560 --> 00:47:18,260 I cannot say that a four-fold axis sits here. 837 00:47:18,260 --> 00:47:20,550 I cannot say a four-fold axis sits here. 838 00:47:20,550 --> 00:47:24,740 I'm gonna have to take each of these combinations in turn and 839 00:47:24,740 --> 00:47:29,250 then step back and see what operations exist about the 840 00:47:29,250 --> 00:47:31,720 various loci within the cell. 841 00:47:31,720 --> 00:47:34,890 So the one that's really different is the combination 842 00:47:34,890 --> 00:47:40,465 of a translation T with the operation a pi over 2. 843 00:47:42,980 --> 00:47:52,300 And my theorem says if I put in a motif and rotate it by 90 844 00:47:52,300 --> 00:47:53,550 degrees to here-- 845 00:47:53,550 --> 00:47:56,770 so this is number 1 and this is number 2-- 846 00:47:56,770 --> 00:48:02,780 and then translate it to a location that's related to the 847 00:48:02,780 --> 00:48:05,010 first by symmetry-- 848 00:48:05,010 --> 00:48:08,030 by translation to get number 3-- 849 00:48:08,030 --> 00:48:15,240 they will be related by an operation b pi over 2 that 850 00:48:15,240 --> 00:48:19,020 sits up along the perpendicular bisector by an 851 00:48:19,020 --> 00:48:25,100 amount x, which is T over 2 times the cotangent of alpha 852 00:48:25,100 --> 00:48:36,483 over 2, T over 2 times the cotangent of 1/2 of pi over 2, 853 00:48:36,483 --> 00:48:37,780 45 degrees. 854 00:48:37,780 --> 00:48:43,660 The cotangent of 45 degrees is unity because the two edges of 855 00:48:43,660 --> 00:48:46,220 the triangle are equal in length. 856 00:48:46,220 --> 00:48:50,860 So this says I should go up along the perpendicular 857 00:48:50,860 --> 00:48:54,760 bisector by an amount that is half the length of the cell. 858 00:48:57,810 --> 00:49:01,410 And that puts me right in the middle of the cell. 859 00:49:01,410 --> 00:49:08,330 So the operation b pi over 2 sits not at the edge of the 860 00:49:08,330 --> 00:49:12,290 cell, but at the center of the cell. 861 00:49:12,290 --> 00:49:17,660 And that, I think you will agree, even in this hastily 862 00:49:17,660 --> 00:49:21,950 sketched diagram, is the way number 1 is related directly 863 00:49:21,950 --> 00:49:23,860 to number 3 in one shot. 864 00:49:26,686 --> 00:49:29,970 Now I could do it on the cheap from here. 865 00:49:29,970 --> 00:49:33,260 But let's show that we have first an 866 00:49:33,260 --> 00:49:35,340 operation a pi over 2. 867 00:49:35,340 --> 00:49:38,520 We've already got the operation c pi. 868 00:49:38,520 --> 00:49:43,070 Let's now combine the operation a minus pi over 2. 869 00:49:46,810 --> 00:49:54,480 And that says that for c minus pi over 2, I would go up the 870 00:49:54,480 --> 00:49:58,270 perpendicular bisector a distance T over 2 times the 871 00:49:58,270 --> 00:50:03,400 cotangent of minus 1/2 of pi over 2. 872 00:50:03,400 --> 00:50:08,010 And the cotangent of a negative angle is minus that. 873 00:50:08,010 --> 00:50:11,520 So I go a distance minus T over 2 along the 874 00:50:11,520 --> 00:50:13,710 perpendicular bisector. 875 00:50:13,710 --> 00:50:19,870 And what that is gonna do is to bring me to the center of 876 00:50:19,870 --> 00:50:22,510 the cell that's directly below. 877 00:50:22,510 --> 00:50:30,000 And this will be the rotation c minus pi over 2. 878 00:50:30,000 --> 00:50:32,650 But everything has to be translation equivalent at the 879 00:50:32,650 --> 00:50:38,460 interval T. So I can just move this operation c minus pi over 880 00:50:38,460 --> 00:50:42,160 2 up to the center of my original cell. 881 00:50:42,160 --> 00:50:45,780 So now I have at the center of the cell all the operations of 882 00:50:45,780 --> 00:50:48,700 a four-fold axis. 883 00:50:48,700 --> 00:50:53,370 So a four-fold axis exists in the middle of the cell. 884 00:50:53,370 --> 00:50:56,660 And let me write the results in here. 885 00:50:56,660 --> 00:50:59,420 I've now got all of the operations of a four-fold axis 886 00:50:59,420 --> 00:51:00,540 sitting here. 887 00:51:00,540 --> 00:51:03,030 I've got all the operations of a two-fold axis 888 00:51:03,030 --> 00:51:04,430 sitting here and here. 889 00:51:04,430 --> 00:51:08,650 So this is the final result, if I translate the two-fold 890 00:51:08,650 --> 00:51:10,950 axes over to the other edges of the cell. 891 00:51:17,410 --> 00:51:18,846 AUDIENCE: What is that angle [? by the d project? ?] 892 00:51:18,846 --> 00:51:21,080 That should be 90 degrees? 893 00:51:21,080 --> 00:51:22,410 PROFESSOR: That should be 90 degrees, 894 00:51:22,410 --> 00:51:23,580 if I drew it carefully. 895 00:51:23,580 --> 00:51:25,090 Yeah. 896 00:51:25,090 --> 00:51:32,250 So let me give the final result with a pattern in it. 897 00:51:32,250 --> 00:51:34,350 These are the two translations, equal in length, 898 00:51:34,350 --> 00:51:37,800 four-fold here, four-fold here, four-fold here. 899 00:51:37,800 --> 00:51:40,620 They are all the same four-fold axis. 900 00:51:40,620 --> 00:51:43,180 Four-fold axis in the middle of the cell. 901 00:51:43,180 --> 00:51:46,970 Two-fold axis in the middle of the cell edges. 902 00:51:46,970 --> 00:51:51,780 And the pattern of objects is the set of things that are 903 00:51:51,780 --> 00:51:56,330 rotated by 90 degrees that form a square about the corner 904 00:51:56,330 --> 00:51:57,580 of the cell. 905 00:51:59,570 --> 00:52:00,960 Another one here. 906 00:52:00,960 --> 00:52:04,760 Another one that sits up here. 907 00:52:04,760 --> 00:52:08,000 And another one that sits like this. 908 00:52:08,000 --> 00:52:11,060 Four of them on the corners of the square. 909 00:52:11,060 --> 00:52:14,830 And that's all we're going to get in this pattern. 910 00:52:14,830 --> 00:52:19,440 Again, it's just a pattern of a four-fold axis that is hung 911 00:52:19,440 --> 00:52:23,970 at every lattice point of the square net. 912 00:52:23,970 --> 00:52:34,010 And these other symmetry elements that arise simply are 913 00:52:34,010 --> 00:52:37,360 things that relate to the squares that are hanging at 914 00:52:37,360 --> 00:52:40,350 every corner of the square cell. 915 00:52:47,820 --> 00:52:48,180 OK. 916 00:52:48,180 --> 00:52:51,970 So the square in the center of the cell, for example, relates 917 00:52:51,970 --> 00:52:53,520 these four. 918 00:52:53,520 --> 00:52:55,030 And you can pick any other four and they'd 919 00:52:55,030 --> 00:52:56,230 be related as well. 920 00:52:56,230 --> 00:52:58,780 The two-fold axis relates this to this. 921 00:52:58,780 --> 00:53:02,490 This two-fold axis relates this to this and this to this. 922 00:53:02,490 --> 00:53:06,910 So the pattern is just the square produced by four 923 00:53:06,910 --> 00:53:09,620 stamped out at every corner of the square net. 924 00:53:09,620 --> 00:53:10,870 And it's that simple. 925 00:53:14,400 --> 00:53:17,164 The name of this thing-- 926 00:53:17,164 --> 00:53:18,080 what have we done? 927 00:53:18,080 --> 00:53:20,740 We've taken a four-fold axis, we've put it 928 00:53:20,740 --> 00:53:22,390 in a primitive net. 929 00:53:22,390 --> 00:53:26,520 I don't have to tell you it's squared because in your heart 930 00:53:26,520 --> 00:53:28,455 of hearts know that that is what a 931 00:53:28,455 --> 00:53:30,480 four-fold axis requires. 932 00:53:30,480 --> 00:53:32,050 So this describes completely the 933 00:53:32,050 --> 00:53:33,810 combinations that's been made. 934 00:53:33,810 --> 00:53:36,010 And this is a representative pattern. 935 00:53:36,010 --> 00:53:38,680 A pattern of this sort is one that we've seen several times 936 00:53:38,680 --> 00:53:40,620 already in describing sample symmetries. 937 00:53:40,620 --> 00:53:44,610 That is the plane group of square floor tiles. 938 00:53:44,610 --> 00:53:50,900 It's the plane group of the square mesh that's in the 939 00:53:50,900 --> 00:53:54,540 overhead lighting fixtures. 940 00:53:54,540 --> 00:53:58,380 It's the pattern that is present in the tiles up above 941 00:53:58,380 --> 00:54:00,060 the lighting fixtures. 942 00:54:00,060 --> 00:54:03,160 It's a pattern that is very commonly present in square 943 00:54:03,160 --> 00:54:06,820 shirts, but not too often. 944 00:54:06,820 --> 00:54:09,070 It's kind of dull and uninteresting. 945 00:54:09,070 --> 00:54:09,890 Yes. 946 00:54:09,890 --> 00:54:13,560 Fellow back there against the wall wearing P4, probably 947 00:54:13,560 --> 00:54:16,410 didn't know until now. 948 00:54:16,410 --> 00:54:17,660 Thank you. 949 00:54:20,030 --> 00:54:21,050 Actually, he's a shield. 950 00:54:21,050 --> 00:54:21,710 I made him come in. 951 00:54:21,710 --> 00:54:23,280 I called him up last night and said be sure 952 00:54:23,280 --> 00:54:24,610 you wear the P4 shirt. 953 00:54:24,610 --> 00:54:27,530 And he said, OK, I will. 954 00:54:27,530 --> 00:54:28,090 All right. 955 00:54:28,090 --> 00:54:28,890 I'm getting silly. 956 00:54:28,890 --> 00:54:31,190 So that's tells me it's time to quit. 957 00:54:31,190 --> 00:54:33,550 So let's take our usual 10 minute break, 958 00:54:33,550 --> 00:54:34,950 And then we'll resume. 959 00:54:34,950 --> 00:54:38,120 And I think I'll go lickety split through the remaining 960 00:54:38,120 --> 00:54:41,610 plane groups that consists of combinations of a rotation 961 00:54:41,610 --> 00:54:42,690 with a lattice. 962 00:54:42,690 --> 00:54:44,580 OK. 963 00:54:44,580 --> 00:54:45,830 So do come back.