1 00:00:08,870 --> 00:00:12,280 PROFESSOR: Any questions about where we left off-- up to 2 00:00:12,280 --> 00:00:13,530 where we left off? 3 00:00:17,050 --> 00:00:21,700 OK, what I'll do then is give you a few more examples of the 4 00:00:21,700 --> 00:00:28,750 combinations in 22 to show which ones we have to retain 5 00:00:28,750 --> 00:00:32,590 as frameworks for crystallographic point groups 6 00:00:32,590 --> 00:00:38,320 and which ones exist as groups but which involve rotational 7 00:00:38,320 --> 00:00:41,080 symmetries that are not permitted to a lattice. 8 00:00:41,080 --> 00:00:46,410 So we've seen a combination of three orthogonal twofold axes 9 00:00:46,410 --> 00:00:50,500 and then projection that would look like this. 10 00:00:50,500 --> 00:00:54,880 And the international symbol for that point group is just a 11 00:00:54,880 --> 00:01:03,190 running list of the different axes that are present, 222. 12 00:01:03,190 --> 00:01:08,940 The next group in the sequence would be 3 2 2, where we took 13 00:01:08,940 --> 00:01:11,940 a 120 degree rotation. 14 00:01:11,940 --> 00:01:15,820 We combine that with a twofold axis perpendicular to it and 15 00:01:15,820 --> 00:01:20,120 the new twofold axis comes out and reminds you again of 16 00:01:20,120 --> 00:01:24,520 things that are quite clear but which are easy to forget-- 17 00:01:24,520 --> 00:01:28,550 that this angle here is 1/2 of 2 pi over 3. 18 00:01:28,550 --> 00:01:29,680 Don't forget that 1/2. 19 00:01:29,680 --> 00:01:34,600 So the neighboring twofold axis is 60 degrees away and 20 00:01:34,600 --> 00:01:43,750 then if we allow these axes to operate on each other, the net 21 00:01:43,750 --> 00:01:47,680 symmetry consistent set of axes looks like this. 22 00:01:47,680 --> 00:01:51,630 But let us look at a solid that has this symmetry. 23 00:01:51,630 --> 00:01:55,350 And such a solid would be a trigonal-- 24 00:01:55,350 --> 00:01:56,600 triangular prism. 25 00:01:59,260 --> 00:02:03,080 And again we can use the corners of this polyhedron as 26 00:02:03,080 --> 00:02:06,440 the reference locations of our motifs. 27 00:02:06,440 --> 00:02:13,170 So let us put a first motif here, number 1. 28 00:02:13,170 --> 00:02:20,500 Let's rotate it by 120 degrees to get a second motif here. 29 00:02:20,500 --> 00:02:25,770 And then let us rotate that one down by a twofold axis 30 00:02:25,770 --> 00:02:27,950 coming out of one of the edges. 31 00:02:27,950 --> 00:02:31,930 That will give us number 3 that is down here or of the 32 00:02:31,930 --> 00:02:33,520 same chirality. 33 00:02:33,520 --> 00:02:37,630 And how do we get from 1 to 3 directly in one shot? 34 00:02:37,630 --> 00:02:41,240 And the answer is about a twofold axis that comes out of 35 00:02:41,240 --> 00:02:42,490 the face of the prism. 36 00:02:45,490 --> 00:02:49,010 So again the prism that we've used as our reference is a 37 00:02:49,010 --> 00:02:52,800 prism that looks like this. 38 00:02:52,800 --> 00:02:55,420 We've got one twofold axis coming out of the face. 39 00:02:55,420 --> 00:03:00,960 The next twofold axis is the one that is equivalent to a 40 00:03:00,960 --> 00:03:03,320 threefold rotation followed by a twofold rotation. 41 00:03:06,460 --> 00:03:10,500 Now here we hit a situation in deciding on the name for the 42 00:03:10,500 --> 00:03:15,040 combination that is analogous to what we found in 43 00:03:15,040 --> 00:03:18,550 two-dimensional plane groups for 3mm. 44 00:03:18,550 --> 00:03:22,170 We saw that only one mirror plane was distinct. 45 00:03:22,170 --> 00:03:25,440 We look at this arrangement of axes-- 46 00:03:25,440 --> 00:03:30,120 this was axis a, alpha, this was axis b, beta. 47 00:03:30,120 --> 00:03:35,880 But if I repeat one of these twofold axes by 120 degree 48 00:03:35,880 --> 00:03:40,130 rotations, this one is the opposite end of this one, this 49 00:03:40,130 --> 00:03:43,000 one is the opposite end of this one, and this one is the 50 00:03:43,000 --> 00:03:44,280 opposite end of this one. 51 00:03:44,280 --> 00:03:47,820 So there are only three kinds of-- there are three, twofold 52 00:03:47,820 --> 00:03:53,080 axes and they are related by the 120 degree rotation. 53 00:03:53,080 --> 00:03:56,680 Another way of saying that is that if we look at a trigonal 54 00:03:56,680 --> 00:04:01,330 prism, each of the twofold axes comes out of an edge and 55 00:04:01,330 --> 00:04:02,490 out of the opposite face. 56 00:04:02,490 --> 00:04:07,560 So there they all do the same thing in this regular prism-- 57 00:04:07,560 --> 00:04:11,130 twofold axes extend between corners of the opposite face. 58 00:04:11,130 --> 00:04:12,330 So there are only-- 59 00:04:12,330 --> 00:04:14,950 there's only one kind of twofold axis present in terms 60 00:04:14,950 --> 00:04:16,790 of being symmetry independent. 61 00:04:16,790 --> 00:04:22,770 So just as we called 3mm, 3m, and we call this one 3 2 62 00:04:22,770 --> 00:04:26,180 because all the twofold axes are symmetry equivalent. 63 00:04:30,030 --> 00:04:33,650 The next one that is crystallographic would be a 64 00:04:33,650 --> 00:04:40,750 combination of a 90 degree rotation with a pair of 65 00:04:40,750 --> 00:04:47,710 twofold axes that are normal to it and separated by 1/2 of 66 00:04:47,710 --> 00:04:50,415 pi over 2, the [? throw ?] of a fourfold axis. 67 00:04:55,490 --> 00:04:58,440 And the symbol for this one would be a fourfold and now 68 00:04:58,440 --> 00:05:01,380 there are two different kinds of twofold axes. 69 00:05:01,380 --> 00:05:07,280 And if we look at a regular square prism, we can again 70 00:05:07,280 --> 00:05:10,070 show that what we've been demonstrating for these other 71 00:05:10,070 --> 00:05:11,750 prisms is true. 72 00:05:11,750 --> 00:05:15,490 One twofold axis would come out of the face, the other 73 00:05:15,490 --> 00:05:18,670 twofold axis would come out of the midpoint of an edge, and 74 00:05:18,670 --> 00:05:21,780 the fourfold axis would come up here. 75 00:05:21,780 --> 00:05:28,030 And a 90 degree rotation, from here to here, followed by a 76 00:05:28,030 --> 00:05:32,320 twofold rotation about this axis, gives us as a net 77 00:05:32,320 --> 00:05:37,370 effect a 1, 2, 3. 78 00:05:37,370 --> 00:05:41,170 And that rotation from 1 to 3 about this twofold axis gives 79 00:05:41,170 --> 00:05:42,420 us the combined mappings. 80 00:05:47,880 --> 00:05:50,790 Notice that the order in which we do them is unimportant. 81 00:05:50,790 --> 00:05:57,500 For example we could do the same thing but do two 180 82 00:05:57,500 --> 00:05:59,670 degree rotations about the twofold axes. 83 00:06:08,260 --> 00:06:11,280 Let's say we start by doing a rotation about 84 00:06:11,280 --> 00:06:12,650 this twofold axis. 85 00:06:12,650 --> 00:06:14,550 From here down to here. 86 00:06:14,550 --> 00:06:18,350 And then we do a twofold axis about the-- 87 00:06:21,490 --> 00:06:24,140 twofold axis that comes out the face and that would take 88 00:06:24,140 --> 00:06:27,940 number 2, and rotate it up to number 3. 89 00:06:27,940 --> 00:06:32,830 And the way we get from 1 to 3 directly is by a rotation of C 90 00:06:32,830 --> 00:06:36,570 pi over 2 about the square face of the prism. 91 00:06:36,570 --> 00:06:38,760 So do them in any order you like-- 92 00:06:38,760 --> 00:06:43,420 two 180 degree rotations, or a 90 degree rotation, one of the 93 00:06:43,420 --> 00:06:46,370 two full rotations with 90 is the other twofold-- 94 00:06:46,370 --> 00:06:48,260 the other type of twofold axis. 95 00:06:48,260 --> 00:06:51,190 The 90 degree rotation plus the second type of the twofold 96 00:06:51,190 --> 00:06:53,230 axis is the same as the first. 97 00:06:53,230 --> 00:06:58,040 So the result can be permuted and turns out to be the same 98 00:06:58,040 --> 00:06:59,290 combination. 99 00:07:02,660 --> 00:07:06,130 The next one that we would hit if we proceed systematically 100 00:07:06,130 --> 00:07:08,500 is non-crystallographic. 101 00:07:08,500 --> 00:07:10,670 And this would be a fivefold axis. 102 00:07:13,470 --> 00:07:17,390 And I'm foolhardy to even start trying to sketch this in 103 00:07:17,390 --> 00:07:18,640 three dimensions. 104 00:07:21,100 --> 00:07:26,570 Nevertheless, nothing ventured, nothing gained. 105 00:07:26,570 --> 00:07:32,190 Start with a twofold axis out of one of the edges and rotate 106 00:07:32,190 --> 00:07:36,750 from 1 down to 2. 107 00:07:36,750 --> 00:07:40,710 Follow that by a twofold rotation about the twofold 108 00:07:40,710 --> 00:07:42,460 axis that comes out of the face. 109 00:07:42,460 --> 00:07:47,870 And that gives us one number 3 up here, and lo and behold, 110 00:07:47,870 --> 00:07:51,070 the way you get from 1 to 3 directly is by a rotation 111 00:07:51,070 --> 00:07:53,810 through 1/5 of 2 pi. 112 00:07:53,810 --> 00:07:57,410 So this would be the non-crystallographic point 113 00:07:57,410 --> 00:08:02,760 group, 5- well it's not going to be 5 2 2. 114 00:08:02,760 --> 00:08:06,020 And that has a whole bunch of twofold axes separated 115 00:08:06,020 --> 00:08:09,020 by 1/10 of 2 pi. 116 00:08:09,020 --> 00:08:14,950 And just as in 3 2, twofold axes here all come out of the 117 00:08:14,950 --> 00:08:17,280 face and out of the opposite edge. 118 00:08:17,280 --> 00:08:19,880 So this would be called 522. 119 00:08:19,880 --> 00:08:22,900 A nice symmetry but not crystallographic, so we can 120 00:08:22,900 --> 00:08:24,150 promptly forget about. 121 00:08:26,980 --> 00:08:31,520 So what comes out of this is a family of groups that are all 122 00:08:31,520 --> 00:08:33,500 of the form n 2 2. 123 00:08:33,500 --> 00:08:40,020 The crystallographic ones are 222, 32, 422 which we've 124 00:08:40,020 --> 00:08:43,549 looked at in detail, and one that I won't draw because 125 00:08:43,549 --> 00:08:46,410 there's so much symmetry it get's messy. 126 00:08:46,410 --> 00:08:53,940 This is 622. 127 00:08:53,940 --> 00:08:57,410 There is a Schoenflies notation. 128 00:08:57,410 --> 00:09:00,590 You remember the language that we encountered for our two 129 00:09:00,590 --> 00:09:03,360 dimensional point groups. 130 00:09:03,360 --> 00:09:07,130 We used m for mirror in the international notation. 131 00:09:07,130 --> 00:09:12,370 We used C standing for cyclic group, subscript s standing 132 00:09:12,370 --> 00:09:15,410 for Spiegel in the Schoenflies notation. 133 00:09:15,410 --> 00:09:18,960 The Schoenflies notation for all of this family of symmetry 134 00:09:18,960 --> 00:09:21,990 is Dn, and the D stands for dihedral. 135 00:09:26,210 --> 00:09:30,310 And the reason for that name is that the difference between 136 00:09:30,310 --> 00:09:34,640 all of these groups, besides the n-fold axis, is this angle 137 00:09:34,640 --> 00:09:38,830 between adjacent twofolds and this is a dihedral angle. 138 00:09:43,560 --> 00:09:47,440 I have a set of planes passing through a common axis; the 139 00:09:47,440 --> 00:09:50,920 angle between those planes is termed a dihedral angle. 140 00:09:50,920 --> 00:09:55,550 So this is called D for dihedral and then a subscript 141 00:09:55,550 --> 00:09:57,200 that gives the rank of the axis. 142 00:09:57,200 --> 00:09:59,190 So Dn generically. 143 00:09:59,190 --> 00:10:05,710 This is D2, this is D3, this is D4, and this is D6. 144 00:10:18,920 --> 00:10:21,270 Comments or debate? 145 00:10:21,270 --> 00:10:22,038 Yes, sir. 146 00:10:22,038 --> 00:10:23,288 AUDIENCE: [INAUDIBLE] 147 00:10:34,938 --> 00:10:37,270 PROFESSOR: [INAUDIBLE] 148 00:10:37,270 --> 00:10:38,420 --out of this face. 149 00:10:38,420 --> 00:10:42,360 So I got from here down to the diametrically opposed axis. 150 00:10:42,360 --> 00:10:44,790 And I rotate it about the adjacent one which comes out 151 00:10:44,790 --> 00:10:46,790 of an edge and-- 152 00:10:46,790 --> 00:10:49,120 what did I do here. 153 00:10:49,120 --> 00:10:57,085 Here I did A pi over 2 from 1 to 2, and then I did B pi, 154 00:10:57,085 --> 00:11:02,400 where this is B pi, and they turned out to be C pi, which 155 00:11:02,400 --> 00:11:04,090 is this one here. 156 00:11:06,650 --> 00:11:10,660 So going from here to here down to number 3 is the same 157 00:11:10,660 --> 00:11:14,010 as going from 1 to 3 in one shot about a twofold axis 158 00:11:14,010 --> 00:11:15,260 normal to the face. 159 00:11:18,290 --> 00:11:21,180 And actually I'm courageous to try to do this in three 160 00:11:21,180 --> 00:11:21,840 dimensions. 161 00:11:21,840 --> 00:11:25,940 We could do it in projection and then things are used-- 162 00:11:25,940 --> 00:11:29,990 this for a point that's up, and use this for a point 163 00:11:29,990 --> 00:11:31,600 that's down. 164 00:11:31,600 --> 00:11:37,830 And then what we've done is to go from 1 that's up, to 2 165 00:11:37,830 --> 00:11:48,200 that's up, and then we rotate it about this twofold axis. 166 00:11:48,200 --> 00:11:51,710 That was 3, that's down. 167 00:11:51,710 --> 00:11:54,990 So when we get into complicated symmetries where 168 00:11:54,990 --> 00:11:58,220 you just can't do a proper job drawing them in an 169 00:11:58,220 --> 00:12:01,190 orthographic drawing, we'll do them in projection and use a 170 00:12:01,190 --> 00:12:04,140 solid dot for something that's up and an open circle for 171 00:12:04,140 --> 00:12:05,390 something that's down. 172 00:12:11,760 --> 00:12:15,410 Is there any other way we can combine things? 173 00:12:15,410 --> 00:12:19,820 Well what you would have to do is use these three relations, 174 00:12:19,820 --> 00:12:22,470 and plug and chug your way through all of the 175 00:12:22,470 --> 00:12:26,080 combinations which were enumerated in the handout. 176 00:12:26,080 --> 00:12:29,020 And I'll save ourselves a lot of work by saying that there 177 00:12:29,020 --> 00:12:32,410 are only two more combinations. 178 00:12:32,410 --> 00:12:37,020 And these are combinations of axes at angles that have 179 00:12:37,020 --> 00:12:40,680 relevance to directions in a cube. 180 00:12:40,680 --> 00:12:44,490 One of them is a twofold axis with a threefold axis with a 181 00:12:44,490 --> 00:12:46,760 threefold rotation. 182 00:12:46,760 --> 00:12:47,830 Again remember these are not 183 00:12:47,830 --> 00:12:49,450 equations in symmetry elements. 184 00:12:49,450 --> 00:12:56,180 This is really A pi, combined with B 2 pi over 3, combined 185 00:12:56,180 --> 00:12:59,810 with C 2 pi over 3. 186 00:12:59,810 --> 00:13:03,320 And the angles that fall out of this are all crazy things 187 00:13:03,320 --> 00:13:08,480 like 109 point something degrees and they make no sense 188 00:13:08,480 --> 00:13:09,130 whatsoever. 189 00:13:09,130 --> 00:13:13,300 They're not nice things like some multiples of 2 pi, 90 190 00:13:13,300 --> 00:13:15,030 degrees or 120 degrees. 191 00:13:15,030 --> 00:13:21,030 And they make no sense at all until you refer them to 192 00:13:21,030 --> 00:13:22,530 directions in a cube. 193 00:13:26,080 --> 00:13:32,360 And this one, 2 3 3, consists of a twofold axis coming out 194 00:13:32,360 --> 00:13:39,090 of the face of the cube, a 120 degree rotation, so this is B 195 00:13:39,090 --> 00:13:43,000 2 pi over 3, this is A pi. 196 00:13:43,000 --> 00:13:49,270 And the other one, C2 pi over 3, corresponds to a threefold 197 00:13:49,270 --> 00:13:51,260 axis coming out of another body diagonal. 198 00:14:04,030 --> 00:14:06,000 I don't know if I want to be gutsy enough to try to 199 00:14:06,000 --> 00:14:08,140 illustrate that that really works. 200 00:14:08,140 --> 00:14:12,540 But one thing that we should do is to let these axes go to 201 00:14:12,540 --> 00:14:16,330 work on one another, and see what comes out. 202 00:14:16,330 --> 00:14:19,650 First thing we can say is that this threefold axis-- 203 00:14:19,650 --> 00:14:22,270 if we extend it, it comes out of the bottom 204 00:14:22,270 --> 00:14:24,360 diagonal of the cube. 205 00:14:24,360 --> 00:14:27,760 This one, if we extend it, will come out of this diagonal 206 00:14:27,760 --> 00:14:28,550 of the cube. 207 00:14:28,550 --> 00:14:31,300 So there's a threefold here, and a threefold here. 208 00:14:31,300 --> 00:14:35,700 The twofold axis gives us a threefold axis that will come 209 00:14:35,700 --> 00:14:38,290 down this way. 210 00:14:38,290 --> 00:14:41,400 So there's a threefold axis here and a threefold axis 211 00:14:41,400 --> 00:14:42,720 coming out here. 212 00:14:42,720 --> 00:14:46,000 And then the twofold axis will rotate this threefold axis 213 00:14:46,000 --> 00:14:49,970 over to the remaining pair of corners. 214 00:14:49,970 --> 00:14:54,940 So we have created this by looking at a twofold rotation, 215 00:14:54,940 --> 00:14:57,490 combined with the threefold axis, combined with the 216 00:14:57,490 --> 00:14:59,790 rotation of another threefold axis. 217 00:14:59,790 --> 00:15:03,990 But in point of fact, if you let the twofold axis operate 218 00:15:03,990 --> 00:15:11,620 on the threefold axis coming out of one body diagonal, you 219 00:15:11,620 --> 00:15:13,980 get threefold axes automatically 220 00:15:13,980 --> 00:15:16,340 out of all body diagonals. 221 00:15:16,340 --> 00:15:20,760 So this is given the international symbol 2 3, 222 00:15:20,760 --> 00:15:23,900 because if you start with one twofold axis out of a face 223 00:15:23,900 --> 00:15:27,420 normal and one threefold axis out of-- 224 00:15:27,420 --> 00:15:31,830 along a body diagonal, the twofold axis, acting on that 225 00:15:31,830 --> 00:15:35,150 threefold axis gives you one along every body diagonal. 226 00:15:35,150 --> 00:15:37,895 And the threefold axis-- 227 00:15:37,895 --> 00:15:43,590 let me point out that a cube standing up on its body 228 00:15:43,590 --> 00:15:48,150 diagonal, with a threefold axis coming out here and these 229 00:15:48,150 --> 00:15:51,030 faces sloping down into the blackboard. 230 00:15:51,030 --> 00:15:54,780 If I have a twofold axis coming out of one face, the 231 00:15:54,780 --> 00:15:58,620 threefold axis puts a twofold axis on this face and rotates 232 00:15:58,620 --> 00:16:02,510 again 120 degrees and puts a twofold axis on this face. 233 00:16:02,510 --> 00:16:06,490 So the threefold axis relates all twofold axes coming out 234 00:16:06,490 --> 00:16:09,790 normal to the cubed face. 235 00:16:09,790 --> 00:16:14,630 And they were really just two independent axes in this 236 00:16:14,630 --> 00:16:16,680 combination. 237 00:16:16,680 --> 00:16:20,160 So 2 3 is the international symbol-- one kind of twofold 238 00:16:20,160 --> 00:16:23,730 axis, one kind of threefold axis-- 239 00:16:23,730 --> 00:16:28,200 inclined at these crazy angles that are in a cube. 240 00:16:28,200 --> 00:16:34,770 The Schoenflies notation for this is T, and that stands for 241 00:16:34,770 --> 00:16:36,020 tetrahedron. 242 00:16:40,510 --> 00:16:47,640 And let me try to convince you that if I look at a 243 00:16:47,640 --> 00:16:52,520 tetrahedron, that that is the arrangement of pure rotation 244 00:16:52,520 --> 00:16:56,070 axes in a tetrahedron. 245 00:16:56,070 --> 00:17:00,110 And the way to show a tetrahedron is to 246 00:17:00,110 --> 00:17:01,650 inscribe it in a cube. 247 00:17:13,119 --> 00:17:18,670 So if I connect these two faces together and these two 248 00:17:18,670 --> 00:17:34,970 faces together, that will define for me a solid that has 249 00:17:34,970 --> 00:17:37,910 four triangular faces. 250 00:17:37,910 --> 00:17:40,705 Easier to recognize it when we put it up on one face. 251 00:17:45,860 --> 00:17:47,110 So this is a tetrahedron. 252 00:17:50,200 --> 00:17:55,540 Schoenflies symbol is T. And I love to get to this part of 253 00:17:55,540 --> 00:18:00,280 the semester and be at this point at the end of the hour, 254 00:18:00,280 --> 00:18:07,000 because if I draw a stereographic projection of 255 00:18:07,000 --> 00:18:15,730 the twofold axes, in this symmetry, I can take a couple 256 00:18:15,730 --> 00:18:19,700 of more twofold axes and add it to this combination which 257 00:18:19,700 --> 00:18:21,660 destroys the group. 258 00:18:21,660 --> 00:18:25,670 But lets me wish everybody a happy Halloween and exit to a 259 00:18:25,670 --> 00:18:28,020 stunned silence at the end of the hour. 260 00:18:32,810 --> 00:18:35,340 So this is a nice point group for October. 261 00:18:39,520 --> 00:18:43,780 There is one more and that is the highest symmetry of all. 262 00:18:43,780 --> 00:18:47,420 And this is a combination of a rotation-- 263 00:18:47,420 --> 00:18:53,200 A pi over 2, a 90 degree rotation, with a rotation B 2 264 00:18:53,200 --> 00:18:57,280 pi over 3, rotation through one third of the circle, and 265 00:18:57,280 --> 00:18:59,630 the rotation C pi. 266 00:18:59,630 --> 00:19:04,650 And the directions between the axes that come out there don't 267 00:19:04,650 --> 00:19:07,050 come out some multiples of 2 pi. 268 00:19:07,050 --> 00:19:10,880 Again they are crazy angles that make no sense at all 269 00:19:10,880 --> 00:19:13,700 unless you refer them to directions 270 00:19:13,700 --> 00:19:16,160 that occur in a cube. 271 00:19:16,160 --> 00:19:19,310 The fourfold axis is in the direction that corresponds to 272 00:19:19,310 --> 00:19:21,180 the normal to a face. 273 00:19:21,180 --> 00:19:24,910 The twofold axis corresponds to a direction out 274 00:19:24,910 --> 00:19:26,300 of one of the edges. 275 00:19:26,300 --> 00:19:29,600 And the threefold axis corresponds to a direction 276 00:19:29,600 --> 00:19:31,940 that is a body diagonal. 277 00:19:31,940 --> 00:19:33,240 So again this is a mess. 278 00:19:33,240 --> 00:19:37,010 If we let those axes work on one another however, we will 279 00:19:37,010 --> 00:19:41,420 produce, more readily appreciated in a stereographic 280 00:19:41,420 --> 00:19:47,310 projection, fourfold axes along the directions that 281 00:19:47,310 --> 00:19:48,710 correspond to face normal. 282 00:19:48,710 --> 00:19:52,620 So the cube, threefold axes coming out of the body 283 00:19:52,620 --> 00:20:00,060 diagonals and twofold axes in between all of the fourfold 284 00:20:00,060 --> 00:20:06,690 axes in directions that correspond to the lines from 285 00:20:06,690 --> 00:20:10,130 the center of the cube out through the edges. 286 00:20:10,130 --> 00:20:14,650 So this is a group which would be called 4 3 2. 287 00:20:14,650 --> 00:20:18,220 There's one kind of fourfold axis related to all the others 288 00:20:18,220 --> 00:20:20,570 by the other rotation axes that are present. 289 00:20:20,570 --> 00:20:23,840 One kind of threefold axis that is related to all of the 290 00:20:23,840 --> 00:20:26,970 other threefold axes along the body diagonal by other 291 00:20:26,970 --> 00:20:28,720 rotation axes that are present. 292 00:20:28,720 --> 00:20:32,410 And twofold axes, one kind, all coming out of the edges. 293 00:20:32,410 --> 00:20:35,070 So this is the group that, in international tables, is 294 00:20:35,070 --> 00:20:36,760 called 4 3 2. 295 00:20:36,760 --> 00:20:41,650 And the Schoenflies notation, this is called O. And that is 296 00:20:41,650 --> 00:20:44,590 the general reaction when one sees this lovely combination. 297 00:20:44,590 --> 00:20:47,980 You go ooh and O is what it's called. 298 00:20:47,980 --> 00:20:51,090 But the O doesn't stand for a gasp, it stands for an 299 00:20:51,090 --> 00:20:52,340 octahedral. 300 00:20:56,080 --> 00:20:59,460 This is the symmetry of an octahedron-- 301 00:20:59,460 --> 00:21:00,910 rotational symmetry of an octahedron. 302 00:21:04,680 --> 00:21:05,290 So that's it. 303 00:21:05,290 --> 00:21:09,280 That is the bestiary of ways in which you can combine 304 00:21:09,280 --> 00:21:13,650 crystallographic rotation axes in space about a fixed point 305 00:21:13,650 --> 00:21:15,220 of intersection. 306 00:21:15,220 --> 00:21:17,360 And there are eleven of them. 307 00:21:17,360 --> 00:21:18,760 There are the axes by themselves-- 308 00:21:18,760 --> 00:21:21,250 1, 2, 3, 4, and 6. 309 00:21:21,250 --> 00:21:29,540 There are the dihedral groups, 222, 32, 422, and 622.. 310 00:21:29,540 --> 00:21:35,690 And then the two cubic groups, T and O. In the international 311 00:21:35,690 --> 00:21:37,670 notation, I'm mixing metaphors. 312 00:21:37,670 --> 00:21:45,970 These are 23 and 432. 313 00:21:45,970 --> 00:21:53,250 I call to your attention the insidious similarity of the 314 00:21:53,250 --> 00:21:59,170 two combinations of axes, 32 and 23. 315 00:21:59,170 --> 00:22:04,740 When the 3 comes first, this is a group of the form n22. 316 00:22:04,740 --> 00:22:09,190 When the 2 comes first, that is the tetrahedral group. 317 00:22:14,910 --> 00:22:19,580 So if you count them all up, there are 4, 2 is 6, and 5. 318 00:22:19,580 --> 00:22:21,945 There are eleven axial combinations. 319 00:22:33,830 --> 00:22:38,450 Quite a few more than the situation in two dimensions 320 00:22:38,450 --> 00:22:42,440 where we just had single rotation axes 1, 2, 3, 4, 6. 321 00:22:42,440 --> 00:22:46,480 Now we have those as in two dimensions but the dihedral 322 00:22:46,480 --> 00:22:51,630 groups and the two cubic arrangements of axes as well. 323 00:22:51,630 --> 00:22:55,410 So we don't have to stretch our vocabulary too much more 324 00:22:55,410 --> 00:22:57,260 to be all inclusive here. 325 00:23:09,460 --> 00:23:11,110 OK, comments? 326 00:23:11,110 --> 00:23:12,460 Takes your breath away, doesn't it? 327 00:23:24,920 --> 00:23:31,160 All right, let me indicate the next step in outline. 328 00:23:40,770 --> 00:23:45,470 What we will next do is introduce our remaining two 329 00:23:45,470 --> 00:23:48,370 symmetry operations into the picture. 330 00:23:48,370 --> 00:23:50,250 We have the eleven axial combinations. 331 00:23:56,150 --> 00:23:59,920 As I said, we can regard these as a framework that we can 332 00:23:59,920 --> 00:24:05,210 decorate with mirror planes and or the inversion center, 333 00:24:05,210 --> 00:24:09,110 which has not appeared until this point because inversion 334 00:24:09,110 --> 00:24:11,620 is inherently a three dimensional transformation. 335 00:24:15,120 --> 00:24:18,300 So what we're going to do is to take these axial 336 00:24:18,300 --> 00:24:21,450 combinations and add another symmetry 337 00:24:21,450 --> 00:24:23,240 operation to the group. 338 00:24:23,240 --> 00:24:25,750 And this as-- 339 00:24:25,750 --> 00:24:28,300 we used the term earlier, this is an extender. 340 00:24:28,300 --> 00:24:31,380 We have something that constitutes a group by itself 341 00:24:31,380 --> 00:24:34,650 then we muck things up by adding another operation. 342 00:24:34,650 --> 00:24:37,580 Remember in all of this we are combining operations, not 343 00:24:37,580 --> 00:24:38,880 symmetry elements. 344 00:24:38,880 --> 00:24:41,840 So we'll take an axial combination and add the 345 00:24:41,840 --> 00:24:45,930 reflection sigma, a reflection operation sigma. 346 00:24:45,930 --> 00:24:52,620 Or take a rotation and combine it with the operation of 347 00:24:52,620 --> 00:24:53,930 inversion as an extender. 348 00:24:57,500 --> 00:25:02,710 So let's itemize the sorts of extenders we should consider. 349 00:25:02,710 --> 00:25:08,090 And the ground rules are that the extender should leave the 350 00:25:08,090 --> 00:25:12,620 arrangement of rotation axes invariant. 351 00:25:12,620 --> 00:25:17,360 Because if it doesn't, we are going to create a rotation 352 00:25:17,360 --> 00:25:23,110 operation that does not conform to the constraints 353 00:25:23,110 --> 00:25:26,270 that we used in Euler's construction. 354 00:25:26,270 --> 00:25:33,200 So for example, if we take 222, which contains the 355 00:25:33,200 --> 00:25:43,690 operations A pi, B pi, C pi and identity, 356 00:25:43,690 --> 00:25:46,810 that's the group 222. 357 00:25:46,810 --> 00:25:52,600 If we would add to the arrangement 222 a mirror plane 358 00:25:52,600 --> 00:25:56,850 that snaked through some arbitrary fashion like this, 359 00:25:56,850 --> 00:26:00,240 that mirror plane is going to reproduce the twofold axis 360 00:26:00,240 --> 00:26:01,970 over to here. 361 00:26:01,970 --> 00:26:07,120 And that is either going to not constitute a group because 362 00:26:07,120 --> 00:26:10,670 this twofold, this twofold, and this twofold don't conform 363 00:26:10,670 --> 00:26:12,480 to Euler's construction. 364 00:26:12,480 --> 00:26:16,150 Or alternatively, if we put it in carefully at 45 degrees 365 00:26:16,150 --> 00:26:19,330 with this twofold axis, we're going to get twofold axes that 366 00:26:19,330 --> 00:26:22,390 are 45 degrees apart and that's going to change this 367 00:26:22,390 --> 00:26:24,460 into a fourfold axis. 368 00:26:24,460 --> 00:26:29,500 So if the addition of a mirror plane does not leave the 369 00:26:29,500 --> 00:26:31,870 arrangement of rotation axes-- 370 00:26:31,870 --> 00:26:33,020 rotation operations-- 371 00:26:33,020 --> 00:26:36,050 invariant, we're either going to get something that's 372 00:26:36,050 --> 00:26:39,160 impossible and does not constitute a group. 373 00:26:39,160 --> 00:26:41,660 Or else we're going to get a combination of rotation 374 00:26:41,660 --> 00:26:44,840 operations of higher symmetry which we've already found 375 00:26:44,840 --> 00:26:48,270 because we went through that process of combination using 376 00:26:48,270 --> 00:26:51,730 Euler's construction in an exhaustive fashion. 377 00:26:51,730 --> 00:26:58,560 So the rule then is that if we add the reflection operation 378 00:26:58,560 --> 00:27:11,410 sigma, then the arrangement of rotation operations must be 379 00:27:11,410 --> 00:27:12,660 left invariant. 380 00:27:40,800 --> 00:27:44,320 OK let's look at the single axes. 381 00:27:44,320 --> 00:27:50,380 If there's an n-fold axis, the ways we can add a reflection 382 00:27:50,380 --> 00:27:57,080 operation to that axis is to pass the reflection operation 383 00:27:57,080 --> 00:27:59,570 through the axis. 384 00:27:59,570 --> 00:28:03,750 And this is going to look very much like the two dimensional 385 00:28:03,750 --> 00:28:09,630 point groups of the form nmm except that rather than having 386 00:28:09,630 --> 00:28:12,940 a mirror line, imagine the whole works as extending 387 00:28:12,940 --> 00:28:16,430 upwards along the rotation axis and space. 388 00:28:16,430 --> 00:28:19,950 So this extender is called a vertical mirror plane. 389 00:28:24,080 --> 00:28:30,350 The other way we could add a mirror plane to an n-fold 390 00:28:30,350 --> 00:28:37,010 rotation axis is to put the mirror plane in an orientation 391 00:28:37,010 --> 00:28:40,090 that's perpendicular to the rotation axis. 392 00:28:40,090 --> 00:28:43,450 That didn't exist in two dimensions because that plane 393 00:28:43,450 --> 00:28:45,240 that's perpendicular to the axis is the 394 00:28:45,240 --> 00:28:46,830 plane of our paper. 395 00:28:46,830 --> 00:28:50,060 And unless we wanted to have a two-sided group that was on 396 00:28:50,060 --> 00:28:53,370 both the top of the paper and the bottom of the paper, and 397 00:28:53,370 --> 00:28:55,890 we make up the rules since it's our ball game. 398 00:28:55,890 --> 00:28:58,590 And that could be a group and these would be the two-sided 399 00:28:58,590 --> 00:29:01,880 plane groups, a plane point groups, but we 400 00:29:01,880 --> 00:29:03,000 didn't do that here. 401 00:29:03,000 --> 00:29:07,215 Adding the mirror plane, the reflection operation sigma, in 402 00:29:07,215 --> 00:29:11,450 a fashion that is normal to the rotation operation, A 2 pi 403 00:29:11,450 --> 00:29:14,450 over n, is another distinct combination. 404 00:29:14,450 --> 00:29:16,865 And this is called, very descriptively a horizontal 405 00:29:16,865 --> 00:29:18,115 mirror plane. 406 00:29:25,640 --> 00:29:31,710 That looks like about all you can do except for the cases 407 00:29:31,710 --> 00:29:41,900 where we have more than one kind of rotation axis present. 408 00:29:41,900 --> 00:29:46,600 So let me use 422 as an example. 409 00:29:46,600 --> 00:29:50,000 We could add a horizontal mirror plane perpendicular to 410 00:29:50,000 --> 00:29:52,330 the principal axis of symmetry, and that would be 411 00:29:52,330 --> 00:29:53,625 the horizontal sigma. 412 00:30:01,760 --> 00:30:05,430 We could add a vertical mirror plane. 413 00:30:05,430 --> 00:30:08,480 And now there are two ways we can do it. 414 00:30:08,480 --> 00:30:12,550 We could put the mirror plane, the operation sigma, through 415 00:30:12,550 --> 00:30:16,620 the fourfold axis and in a fashion that was perpendicular 416 00:30:16,620 --> 00:30:18,110 to the twofold axis. 417 00:30:18,110 --> 00:30:21,400 And we will retain the term of vertical 418 00:30:21,400 --> 00:30:24,370 sigma for that addition. 419 00:30:24,370 --> 00:30:26,980 But the other thing that we could do would be to put the 420 00:30:26,980 --> 00:30:31,140 reflection operation interleaved between the 421 00:30:31,140 --> 00:30:33,900 twofold axes. 422 00:30:33,900 --> 00:30:36,390 That's going to take this one and flip it into this one, 423 00:30:36,390 --> 00:30:38,550 this one flip it into this one, flip these back and 424 00:30:38,550 --> 00:30:41,300 forth, and that doesn't create any new reflection. 425 00:30:41,300 --> 00:30:50,100 And this is referred to as a diagonal reflection plane. 426 00:30:50,100 --> 00:30:51,280 Diagonal to what? 427 00:30:51,280 --> 00:30:56,710 Diagonally interleaved between the twofold axes. 428 00:30:56,710 --> 00:30:59,280 And these are distinct additions and they will lead 429 00:30:59,280 --> 00:31:00,530 to different groups. 430 00:31:04,448 --> 00:31:09,120 In as far as addition of reflection operations is 431 00:31:09,120 --> 00:31:13,620 concerned, that's about all we can do that's distinct. 432 00:31:13,620 --> 00:31:20,810 And notice that this is for the groups Dn, and 433 00:31:20,810 --> 00:31:23,570 tetrahedral, and octahedral only. 434 00:31:23,570 --> 00:31:27,250 The distinction here is not defined for 435 00:31:27,250 --> 00:31:28,500 just a single axis. 436 00:31:30,930 --> 00:31:35,010 So there are three possible extenders here-- a vertical 437 00:31:35,010 --> 00:31:37,800 mirror plane, a horizontal mirror plane, a diagonal 438 00:31:37,800 --> 00:31:41,920 mirror plane, added to each of the eleven arrangements of 439 00:31:41,920 --> 00:31:44,640 rotation axes. 440 00:31:44,640 --> 00:31:46,860 And then the final extender that we could 441 00:31:46,860 --> 00:31:49,025 add is to add inversion. 442 00:31:55,510 --> 00:31:59,390 And the symbol for the inversion operation is 1 bar. 443 00:31:59,390 --> 00:32:04,200 And obviously if one point in space is going to be left 444 00:32:04,200 --> 00:32:14,800 invariant, you either add this on a single axis, and that is 445 00:32:14,800 --> 00:32:22,320 what you'd have to do for the groups Cn, or at the point of 446 00:32:22,320 --> 00:32:23,570 intersection. 447 00:32:33,920 --> 00:32:45,620 And that would be the case if more than one axis, and that's 448 00:32:45,620 --> 00:32:49,770 the groups of the form n22, T, 449 00:32:49,770 --> 00:32:54,912 and O. That's it. 450 00:32:54,912 --> 00:32:57,580 That's the job. 451 00:32:57,580 --> 00:33:02,070 So we should consider each of these possible additions of an 452 00:33:02,070 --> 00:33:04,210 extender systematically. 453 00:33:04,210 --> 00:33:08,460 I don't propose to do every single one independently. 454 00:33:08,460 --> 00:33:10,790 If we do a couple, you'll get the general idea. 455 00:33:10,790 --> 00:33:13,160 And I think because I have an honest face, and you've come 456 00:33:13,160 --> 00:33:16,010 to trust me, I can just describe the remaining results 457 00:33:16,010 --> 00:33:18,900 to you and we won't grind through every single one. 458 00:33:27,300 --> 00:33:31,950 Now the enormity of what I've proposed becomes apparent when 459 00:33:31,950 --> 00:33:36,640 I say that we now are going to have need of a number of 460 00:33:36,640 --> 00:33:37,870 different-- 461 00:33:37,870 --> 00:33:41,630 what I call combination theorems, that let us complete 462 00:33:41,630 --> 00:33:43,850 the group multiplication table. 463 00:33:43,850 --> 00:33:49,290 And deduce, as a consequence, which symmetry operations must 464 00:33:49,290 --> 00:33:52,310 come into being because of these additions. 465 00:33:52,310 --> 00:33:55,790 So we'll want to know what happens when you add a 466 00:33:55,790 --> 00:34:01,370 vertical sigma to a rotation operation, A 2 pi over n. 467 00:34:01,370 --> 00:34:03,330 We'll want to know what happens when you add a 468 00:34:03,330 --> 00:34:08,650 horizontal sigma to a rotation operation, A 2 pi over n. 469 00:34:08,650 --> 00:34:11,330 And we're going to want to know what happens when you add 470 00:34:11,330 --> 00:34:16,719 a diagonal mirror plane, this really is a special case of 471 00:34:16,719 --> 00:34:19,340 the vertical mirror plane. 472 00:34:19,340 --> 00:34:21,280 And we'll want to know what happens when you add an 473 00:34:21,280 --> 00:34:24,089 inversion center to a rotation axis. 474 00:34:27,670 --> 00:34:31,739 So let me do a few of these and then next time we can 475 00:34:31,739 --> 00:34:34,969 start off and start driving the three dimensional 476 00:34:34,969 --> 00:34:36,219 symmetries. 477 00:34:52,420 --> 00:34:58,920 So let's just repeat the ones that we've already done. 478 00:34:58,920 --> 00:35:04,230 We said that if we have a rotation operation A alpha, 479 00:35:04,230 --> 00:35:10,720 and we put a reflection plane through it, that A alpha 480 00:35:10,720 --> 00:35:13,710 followed by a reflection plane passing through it-- let me 481 00:35:13,710 --> 00:35:16,430 call this sigma V-- because this is the so-called 482 00:35:16,430 --> 00:35:18,440 vertical, up orientation. 483 00:35:18,440 --> 00:35:20,580 We've already seen that in two dimensions. 484 00:35:20,580 --> 00:35:24,650 This is a vertical mirror plane, sigma prime, that is 485 00:35:24,650 --> 00:35:29,330 going to be alpha over 2 away from the first. 486 00:35:29,330 --> 00:35:33,200 So that is something that we've already seen in it's 487 00:35:33,200 --> 00:35:36,290 entirety in the two dimensional point groups. 488 00:35:36,290 --> 00:35:46,170 There was m sigma, there was 2mm, and that was C2V, 3m, 489 00:35:46,170 --> 00:35:57,020 that's C3V, 4mm, and that was C6V, and 6mm, and that's C4V, 490 00:35:57,020 --> 00:35:59,510 and that's C6V. 491 00:35:59,510 --> 00:36:02,970 So that is the result that we obtained for two dimensions. 492 00:36:02,970 --> 00:36:06,130 And you can see now the reason for distinguishing the mirror 493 00:36:06,130 --> 00:36:09,810 plane by saying it is a vertical mirror plane because 494 00:36:09,810 --> 00:36:12,530 this is in the three dimensional sense. 495 00:36:12,530 --> 00:36:16,130 It's vertical parallel to and passing through the rotation 496 00:36:16,130 --> 00:36:19,035 axis-- no longer a rotation point but a rotation axis. 497 00:36:21,680 --> 00:36:22,930 So we've got those theorems. 498 00:36:27,140 --> 00:36:36,780 What happens if we take a rotation operation, A pi, and 499 00:36:36,780 --> 00:36:42,590 put a horizontal reflection operation through it 500 00:36:42,590 --> 00:36:44,230 normal to that axis? 501 00:36:44,230 --> 00:36:47,635 So I'll call this sigma h, a horizontal operation. 502 00:36:52,640 --> 00:36:57,720 OK what we have to do is draw it out once and for all. 503 00:36:57,720 --> 00:37:00,500 Here's the first one, let's say it's right-handed. 504 00:37:00,500 --> 00:37:04,540 We'll rotate by A pi to get a second one which stays 505 00:37:04,540 --> 00:37:05,620 right-handed. 506 00:37:05,620 --> 00:37:10,210 And then we'll reflect it down in the horizontal mirror plane 507 00:37:10,210 --> 00:37:14,260 to get a third one which is left-handed. 508 00:37:14,260 --> 00:37:19,550 And now the question is, how is number one related to 509 00:37:19,550 --> 00:37:22,640 number three? 510 00:37:22,640 --> 00:37:23,970 Anybody want to hazard a guess? 511 00:37:23,970 --> 00:37:25,730 We've got to go from a right-handed one to a 512 00:37:25,730 --> 00:37:27,590 left-handed one. 513 00:37:27,590 --> 00:37:29,400 But these two guys are oriented 514 00:37:29,400 --> 00:37:31,480 anti-parallel to one another. 515 00:37:31,480 --> 00:37:34,830 So how do we relate the first one to the third one? 516 00:37:38,410 --> 00:37:43,560 I heard somebody mumble softly enough to remain anonymous. 517 00:37:43,560 --> 00:37:45,651 Inversion. 518 00:37:45,651 --> 00:37:47,560 Right. 519 00:37:47,560 --> 00:37:51,830 So as we go along making these combinations, if we had not 520 00:37:51,830 --> 00:37:54,990 been bright enough to think of the operation of inversion as 521 00:37:54,990 --> 00:37:58,120 a general transformation where the sense of all three 522 00:37:58,120 --> 00:38:00,880 coordinates is changed, we would have stumbled over 523 00:38:00,880 --> 00:38:02,470 headlong right here. 524 00:38:02,470 --> 00:38:08,280 Combine a rotation operation, A pi, with a mirror reflection 525 00:38:08,280 --> 00:38:10,210 that is perpendicular to the axis. 526 00:38:10,210 --> 00:38:13,680 The way you get from one to three in one shot is by 527 00:38:13,680 --> 00:38:19,510 inversion through a point that is at the intersection between 528 00:38:19,510 --> 00:38:22,610 the rotation axis and the mirror plane. 529 00:38:22,610 --> 00:38:30,500 So rotation followed by rotation in a vertical mirror 530 00:38:30,500 --> 00:38:34,640 plane that's perpendicular to the axis is the operation of 531 00:38:34,640 --> 00:38:36,545 inversion at the point of intersection. 532 00:38:43,260 --> 00:38:47,070 So again, if we had not been clever enough to invent it or 533 00:38:47,070 --> 00:38:49,320 tell you about in advance, there it is. 534 00:38:49,320 --> 00:38:52,750 When we start forming the group multiplication table, we 535 00:38:52,750 --> 00:38:55,800 would have had to have defined this operation to describe 536 00:38:55,800 --> 00:38:57,514 this relation. 537 00:38:57,514 --> 00:39:00,280 AUDIENCE: Is that sigma sub h? 538 00:39:00,280 --> 00:39:01,672 PROFESSOR: Ah yeah. 539 00:39:01,672 --> 00:39:04,010 Let me write that, sigma h. 540 00:39:04,010 --> 00:39:06,795 I thought that V didn't look like a V, so I changed it so 541 00:39:06,795 --> 00:39:08,500 it looked like a V but it shouldn't be a V. That's a 542 00:39:08,500 --> 00:39:09,750 horizontal mirror plane. 543 00:39:21,970 --> 00:39:31,990 OK this is a new combination and if we see what operations 544 00:39:31,990 --> 00:39:35,080 are going to be present in the group, we've got the two 545 00:39:35,080 --> 00:39:39,930 operations of the twofold axis, 1 and A pi. 546 00:39:39,930 --> 00:39:46,230 And what we have added is sigma h as an extender. 547 00:39:46,230 --> 00:39:53,910 So 1 and A pi, this little box here is the subgroup that we 548 00:39:53,910 --> 00:39:56,470 know and love as the twofold axis. 549 00:39:56,470 --> 00:39:59,680 And then we'll write sigma h here and now let's fill in the 550 00:39:59,680 --> 00:40:00,970 group multiplication table. 551 00:40:00,970 --> 00:40:04,840 Doing the identity operation twice is identity. 552 00:40:04,840 --> 00:40:08,020 Doing the identity operation followed by A pi is A pi. 553 00:40:08,020 --> 00:40:12,830 Identity followed by sigma h is sigma h. 554 00:40:12,830 --> 00:40:17,380 Identity followed by A pi, sigma h, lets me 555 00:40:17,380 --> 00:40:18,830 fill in those boxes. 556 00:40:18,830 --> 00:40:22,470 Do a rotation twice, that's the identity operation. 557 00:40:22,470 --> 00:40:26,670 Do a rotation of A pi and follow that by a reflection, A 558 00:40:26,670 --> 00:40:29,260 pi followed by reflection. 559 00:40:29,260 --> 00:40:33,580 This is the inversion operation. 560 00:40:33,580 --> 00:40:37,630 And so I should add inversion to my list of operations since 561 00:40:37,630 --> 00:40:38,880 it's come up. 562 00:40:41,420 --> 00:40:44,950 So 1 followed by inversion is inversion. 563 00:40:44,950 --> 00:40:50,135 A pi followed by inversion is the horizontal reflection. 564 00:40:53,890 --> 00:40:58,060 Horizontal reflection followed by A pi 565 00:40:58,060 --> 00:40:59,310 is the same as inversion. 566 00:41:01,810 --> 00:41:07,530 Horizontal reflection followed by horizontal reflection is 567 00:41:07,530 --> 00:41:08,720 the identity operation-- 568 00:41:08,720 --> 00:41:10,220 brings me back to where I started from. 569 00:41:10,220 --> 00:41:13,770 And a horizontal reflection followed by inversion is the 570 00:41:13,770 --> 00:41:16,310 same as A pi. 571 00:41:16,310 --> 00:41:21,330 So I'll have another object down here to complete the 572 00:41:21,330 --> 00:41:23,730 three dimensional arrangement. 573 00:41:23,730 --> 00:41:26,980 And this is the group for operations-- 574 00:41:26,980 --> 00:41:29,920 A pi, inversion, horizontal reflection, and 575 00:41:29,920 --> 00:41:31,170 the identity operation. 576 00:41:36,340 --> 00:41:40,180 So what do we call this one? 577 00:41:40,180 --> 00:41:45,920 The operation A pi plus a horizontal reflection 578 00:41:45,920 --> 00:41:51,410 operation gives rise to a group that is a twofold axis 579 00:41:51,410 --> 00:41:53,510 of its operations, 580 00:41:53,510 --> 00:41:55,710 perpendicular to a mirror plane. 581 00:41:55,710 --> 00:41:59,770 And this written as a fraction means that the 2 is 582 00:41:59,770 --> 00:42:02,620 perpendicular to the mirror plane rather than being 583 00:42:02,620 --> 00:42:05,760 parallel and in the plane of the mirror plane. 584 00:42:05,760 --> 00:42:07,880 So two codes for writing symbols. 585 00:42:07,880 --> 00:42:12,090 A 2 followed on the same line by the m means that the m is 586 00:42:12,090 --> 00:42:13,070 parallel to 2. 587 00:42:13,070 --> 00:42:15,810 We know that when we write them as a fraction, that will 588 00:42:15,810 --> 00:42:18,740 be our way of designating that this mirror plane is 589 00:42:18,740 --> 00:42:20,660 perpendicular to a twofold axis. 590 00:42:27,350 --> 00:42:32,230 OK it's the witching hour, 4 o'clock exactly. 591 00:42:32,230 --> 00:42:35,440 That's when we ought to quit, so let's stop there.