1 00:00:07,600 --> 00:00:10,710 PROFESSOR: We are thundering rapidly to the 2 00:00:10,710 --> 00:00:12,620 midpoint of the term. 3 00:00:12,620 --> 00:00:18,720 I think a week from this coming Thursday will be 4 00:00:18,720 --> 00:00:23,020 exactly halfway, and the halfway point is when I 5 00:00:23,020 --> 00:00:27,380 usually try, give or take a day, to change gears very 6 00:00:27,380 --> 00:00:31,080 abruptly and start looking at the symmetry of physical 7 00:00:31,080 --> 00:00:35,980 properties and tensors and actual discussion of physical 8 00:00:35,980 --> 00:00:38,100 properties that are anisotropic. 9 00:00:38,100 --> 00:00:39,860 So we'll be finishing up our discussion of 10 00:00:39,860 --> 00:00:42,370 symmetry fairly quickly. 11 00:00:42,370 --> 00:00:47,520 We still have, however, some major derivations to perform, 12 00:00:47,520 --> 00:00:51,430 and we'll get into one of these today. 13 00:00:51,430 --> 00:00:56,810 Last time we had begun the process of deriving what are 14 00:00:56,810 --> 00:01:06,220 called the point groups as opposed to the plane groups. 15 00:01:06,220 --> 00:01:09,220 The plane groups were combinations of symmetry with 16 00:01:09,220 --> 00:01:12,950 a lattice that extended throughout a plane in space. 17 00:01:12,950 --> 00:01:16,510 A point group we've seen in the form of two-dimensional 18 00:01:16,510 --> 00:01:17,340 point groups. 19 00:01:17,340 --> 00:01:20,620 We're now going to today derive the three-dimensional 20 00:01:20,620 --> 00:01:21,630 point groups. 21 00:01:21,630 --> 00:01:23,920 And, again, the name stems from the fact that these are 22 00:01:23,920 --> 00:01:28,710 clusters of symmetry about a common point so that at very 23 00:01:28,710 --> 00:01:32,250 least that one point in space stays put. 24 00:01:32,250 --> 00:01:33,770 So, hence, point groups-- 25 00:01:33,770 --> 00:01:35,800 these are symmetries about a point. 26 00:01:35,800 --> 00:01:41,150 And to put some embellishing adjectives on here, these will 27 00:01:41,150 --> 00:01:47,670 be the three-dimensional crystallographic point groups. 28 00:01:47,670 --> 00:01:50,190 Because there are an infinite number of point groups, we're 29 00:01:50,190 --> 00:01:52,930 considering just those symmetries that can be 30 00:01:52,930 --> 00:01:55,340 combined with a lattice and therefore are 31 00:01:55,340 --> 00:01:57,970 permissible to crystals. 32 00:01:57,970 --> 00:02:01,530 I'd like to point out in case you've not been making much 33 00:02:01,530 --> 00:02:06,580 use of Buerger's book that we actually are getting back to 34 00:02:06,580 --> 00:02:07,570 Buerger's treatment. 35 00:02:07,570 --> 00:02:10,759 We'll do some of the derivations over the next few 36 00:02:10,759 --> 00:02:15,810 days with a little bit more detail and with a slightly 37 00:02:15,810 --> 00:02:17,420 different procedure than Buerger. 38 00:02:17,420 --> 00:02:21,680 But the bases covered are the same. 39 00:02:21,680 --> 00:02:24,920 So let me just point out where some of this material is in 40 00:02:24,920 --> 00:02:26,790 Buerger's book. 41 00:02:26,790 --> 00:02:34,420 Buerger does Euler's Construction, and he does this 42 00:02:34,420 --> 00:02:35,540 in a once over lightly. 43 00:02:35,540 --> 00:02:37,660 I think this is not the best part of the book. 44 00:02:37,660 --> 00:02:43,080 This is on pages 35 to 43. 45 00:02:43,080 --> 00:02:46,730 In particular, he sort of leaps ahead quickly and just 46 00:02:46,730 --> 00:02:51,740 looks for one angle between rotation axes and doesn't get 47 00:02:51,740 --> 00:02:54,860 the-- do specifically the ones that are 90 degrees and are 48 00:02:54,860 --> 00:02:56,620 not as interesting. 49 00:02:56,620 --> 00:03:03,860 Some of the combination theorems are in Chapter 6, and 50 00:03:03,860 --> 00:03:12,300 there you'll find the statement that two reflections 51 00:03:12,300 --> 00:03:18,600 at an angle mu is equivalent to a net rotation of mu, 52 00:03:18,600 --> 00:03:20,600 except Buerger doesn't call them sigmas. 53 00:03:20,600 --> 00:03:21,640 He calls them m's. 54 00:03:21,640 --> 00:03:25,070 He uses the same symbol for individual operations and for 55 00:03:25,070 --> 00:03:29,500 the symmetry of them, and I think that's a little untidy. 56 00:03:29,500 --> 00:03:34,470 Also, the theorem that says that if we have a mirror plane 57 00:03:34,470 --> 00:03:38,360 that's perpendicular to a twofold rotation axis, that 58 00:03:38,360 --> 00:03:43,280 gives rise to inversion center at the point of intersection. 59 00:03:43,280 --> 00:03:47,210 And then he launches into a rather condensed version of 60 00:03:47,210 --> 00:03:53,180 the derivation of the point groups on pages 59 to 68. 61 00:03:56,640 --> 00:04:05,905 Then, in Chapter 7, derives the two dimensional lattices 62 00:04:05,905 --> 00:04:07,740 or the plane nets. 63 00:04:07,740 --> 00:04:11,330 This is on page 69 to 83. 64 00:04:11,330 --> 00:04:14,520 We've done that differently and much more exhaustively by 65 00:04:14,520 --> 00:04:18,100 deriving the two-dimensional plane groups and the lattices 66 00:04:18,100 --> 00:04:20,440 are part of the plane groups, and we've gotten them 67 00:04:20,440 --> 00:04:22,240 automatically. 68 00:04:22,240 --> 00:04:25,550 There is a place where in the derivation of the space 69 00:04:25,550 --> 00:04:28,190 lattices, he's a little bit incomplete and actually does 70 00:04:28,190 --> 00:04:29,480 something that's wrong A-ha-- 71 00:04:29,480 --> 00:04:33,400 I showed Buerger was wrong at one place in his book. 72 00:04:33,400 --> 00:04:36,020 But anyway, this is where the material that we're covering 73 00:04:36,020 --> 00:04:39,730 now is mentioned in Buerger's book. 74 00:04:39,730 --> 00:04:45,050 Last time, we had hit a new surprise. 75 00:04:45,050 --> 00:04:51,230 We had asked if we take the basic rotational symmetries 76 00:04:51,230 --> 00:04:56,250 and add an inversion to something like three-- 77 00:04:56,250 --> 00:05:01,230 so, three is a subgroup, the operation of 78 00:05:01,230 --> 00:05:02,760 inversion is an extender. 79 00:05:06,640 --> 00:05:11,560 We found that there was a new operation that came up, and 80 00:05:11,560 --> 00:05:16,805 this was a rotoinversion operation. 81 00:05:22,200 --> 00:05:25,870 And then the resulting point group, which we called 3-bar-- 82 00:05:28,380 --> 00:05:32,090 the symbol itself, 3-bar, is a threefold rotoinversion. 83 00:05:39,700 --> 00:05:42,180 And if we were not clever enough to invent a 84 00:05:42,180 --> 00:05:46,040 rotoinversion operation, we would have stumbled headlong 85 00:05:46,040 --> 00:05:48,760 over it in adding inversion as an extender 86 00:05:48,760 --> 00:05:51,170 to a threefold axis. 87 00:05:51,170 --> 00:05:55,720 And a rotoinversion axis is something that involves in 88 00:05:55,720 --> 00:06:00,030 general rotating through some angle alpha to get from a 89 00:06:00,030 --> 00:06:05,360 first object to a second object of the same handedness 90 00:06:05,360 --> 00:06:10,310 and then not yet putting it down, first inverting it 91 00:06:10,310 --> 00:06:11,480 through axis-- 92 00:06:11,480 --> 00:06:13,050 through a point on the axis-- 93 00:06:13,050 --> 00:06:15,520 to get a second object. 94 00:06:15,520 --> 00:06:19,080 I shouldn't call this second because this is not rotation 95 00:06:19,080 --> 00:06:22,530 with inversion simultaneously but a two step operation. 96 00:06:22,530 --> 00:06:27,450 So we rotate from 1 to this virtual object also of same 97 00:06:27,450 --> 00:06:30,880 handedness and then before putting it down, we invert it 98 00:06:30,880 --> 00:06:33,620 to get number two, which is left handed. 99 00:06:33,620 --> 00:06:37,020 And the operation of getting there in two steps but in one 100 00:06:37,020 --> 00:06:41,920 repetition is the operation of rotoinversion. 101 00:06:41,920 --> 00:06:46,280 That operation was one of the members of the group that 102 00:06:46,280 --> 00:06:50,360 results when you add inversion as an extender to 3. 103 00:06:50,360 --> 00:06:55,570 But, really, this was not something that was a new sort 104 00:06:55,570 --> 00:06:56,465 of symmetry element. 105 00:06:56,465 --> 00:06:59,960 It was an operation that came up, but we could regard the 106 00:06:59,960 --> 00:07:03,040 point down that we've labeled 3-bar as simply a threefold 107 00:07:03,040 --> 00:07:05,910 axis with an inversion center sitting on it. 108 00:07:05,910 --> 00:07:08,100 But then we looked at another one. 109 00:07:08,100 --> 00:07:11,670 And by anticipation, I said we ought to look at these. 110 00:07:11,670 --> 00:07:17,640 And we looked at what happened when we took a fourfold 111 00:07:17,640 --> 00:07:21,780 rotoinversion operation where we would take the first object 112 00:07:21,780 --> 00:07:25,060 number one, rotate it through 90 degrees, and 113 00:07:25,060 --> 00:07:26,410 not yet put it down. 114 00:07:26,410 --> 00:07:30,390 We'd first invert it through a point on an axis, and we got 115 00:07:30,390 --> 00:07:34,090 one that was 90 degrees away but pointing down in of 116 00:07:34,090 --> 00:07:36,510 opposite handedness. 117 00:07:36,510 --> 00:07:41,370 If we repeated that operation several times, we found that 118 00:07:41,370 --> 00:07:45,740 we got two objects up above a plane normal to the rotation 119 00:07:45,740 --> 00:07:48,860 part of the operation and passing through the point, 120 00:07:48,860 --> 00:07:49,810 which we inverted. 121 00:07:49,810 --> 00:07:52,780 So there's one right handed one up here, a second right 122 00:07:52,780 --> 00:07:57,830 handed one up here, and then two down below, which were of 123 00:07:57,830 --> 00:08:02,870 opposite chirality, number 3 and 4. 124 00:08:02,870 --> 00:08:05,010 And that is an entirely new beast. 125 00:08:05,010 --> 00:08:08,510 We have no way of describing the relation between these 126 00:08:08,510 --> 00:08:12,230 four objects other than saying rotate, don't put it down, 127 00:08:12,230 --> 00:08:13,670 first invert. 128 00:08:13,670 --> 00:08:17,070 Rotate, don't put it down, first invert again. 129 00:08:17,070 --> 00:08:19,370 So this is something entirely new. 130 00:08:19,370 --> 00:08:21,200 It's a two step operation. 131 00:08:21,200 --> 00:08:24,780 You cannot describe it any more simply than saying take 132 00:08:24,780 --> 00:08:28,660 two steps to do the repetition in the same sense that the 133 00:08:28,660 --> 00:08:32,220 glide plane that we discovered in the plane groups had a step 134 00:08:32,220 --> 00:08:36,200 of translation followed immediately by reflection, and 135 00:08:36,200 --> 00:08:38,120 that was a new sort of operation. 136 00:08:38,120 --> 00:08:41,630 So here's another example of a two-step operation. 137 00:08:41,630 --> 00:08:45,110 The symbol for this point group is 4-bar, and the symbol 138 00:08:45,110 --> 00:08:49,140 for it-- if you want to indicate its locus, is, it's 139 00:08:49,140 --> 00:08:53,450 got a squareness to it because these objects as mutually are 140 00:08:53,450 --> 00:08:56,630 separated by 90 degrees, but it really has only a twofold 141 00:08:56,630 --> 00:09:00,310 symmetry, so the symmetry for a 4-bar axis. 142 00:09:00,310 --> 00:09:01,790 It's not really an axis at all. 143 00:09:01,790 --> 00:09:03,840 It's a 4-bar point, because only a 144 00:09:03,840 --> 00:09:05,660 point is left invariant. 145 00:09:05,660 --> 00:09:10,390 But the symbol for a 4-bar "axis" is this square with a 146 00:09:10,390 --> 00:09:14,270 little twofold axis inscribed in it. 147 00:09:14,270 --> 00:09:14,940 Alright. 148 00:09:14,940 --> 00:09:19,090 We ought to really, then, step back and ask how many 149 00:09:19,090 --> 00:09:24,540 different rotoinversion axes there might be, and add these 150 00:09:24,540 --> 00:09:26,170 to our bags of tricks. 151 00:09:26,170 --> 00:09:30,350 And what I'm going to invite you to do is to examine this 152 00:09:30,350 --> 00:09:34,840 systematically on a problem set which I will pass around, 153 00:09:34,840 --> 00:09:36,850 along with several other problems for your 154 00:09:36,850 --> 00:09:38,100 consideration, as well. 155 00:09:43,820 --> 00:09:47,770 And having stumbled onto that combination, another thing we 156 00:09:47,770 --> 00:09:53,350 might ask to broaden our bag of tricks still further is to 157 00:09:53,350 --> 00:09:56,665 say is there such a thing as a rotoreflection axis? 158 00:10:04,500 --> 00:10:08,240 And we called n-bar a rotoinversion. 159 00:10:11,260 --> 00:10:16,140 I wrote a reflection axis is indicated by an n with this 160 00:10:16,140 --> 00:10:17,590 little squiggle on top. 161 00:10:17,590 --> 00:10:21,610 Anyone who has studied Spanish realizes that the proper name 162 00:10:21,610 --> 00:10:28,240 for the squiggle is a tilde, T-I-L-D-E. 163 00:10:28,240 --> 00:10:31,480 And this would be an operation where there's a locus in which 164 00:10:31,480 --> 00:10:35,130 we'll perform the reflection part of the operation, so this 165 00:10:35,130 --> 00:10:38,550 would be a 2 step operation that involves taking a first 166 00:10:38,550 --> 00:10:43,080 one and rotating it to a virtual object number two but 167 00:10:43,080 --> 00:10:44,490 not putting it down. 168 00:10:44,490 --> 00:10:46,500 Before putting it down, we will 169 00:10:46,500 --> 00:10:49,080 reflect in a planar locus. 170 00:10:49,080 --> 00:10:52,270 So this is number two, and it will be of opposite chirality. 171 00:10:55,290 --> 00:10:58,550 So that's another operation that involves rotation coupled 172 00:10:58,550 --> 00:11:02,020 as one part of a step that involves R, the second of our 173 00:11:02,020 --> 00:11:06,410 three translation free operations, namely reflection 174 00:11:06,410 --> 00:11:07,710 and inversion. 175 00:11:07,710 --> 00:11:11,830 So are these something we should consider? 176 00:11:11,830 --> 00:11:16,820 What I've invited you to do is to draw out for n equals 1 to 177 00:11:16,820 --> 00:11:26,700 8 the patterns produced by rotoinversion and the patterns 178 00:11:26,700 --> 00:11:31,220 created by rotoreflection, and then see if you regard these 179 00:11:31,220 --> 00:11:34,830 as new operations or whether they can be broken down into 180 00:11:34,830 --> 00:11:37,530 the simultaneous presence of more than one of our old 181 00:11:37,530 --> 00:11:41,970 friends such as 3-bar being a plain old threefold axis with 182 00:11:41,970 --> 00:11:43,560 an inversion sitting on it. 183 00:11:48,380 --> 00:11:50,940 The fact that I'm asking you take your time to do this 184 00:11:50,940 --> 00:11:53,410 suggests that yes, you are going to find some 185 00:11:53,410 --> 00:11:56,970 rotoinversion and rotoreflection axes, which are 186 00:11:56,970 --> 00:11:59,790 inherently 2 step operations and can't be decomposed. 187 00:12:05,800 --> 00:12:13,110 What I'd like to do now, then, is to systematically look at 188 00:12:13,110 --> 00:12:17,100 the axial combinations. 189 00:12:17,100 --> 00:12:26,548 These are of the form n one, two, three, four, and six and 190 00:12:26,548 --> 00:12:29,220 the dihedral symmetries. 191 00:12:29,220 --> 00:12:34,780 These are of the form n22, or just 32 in the case of the one 192 00:12:34,780 --> 00:12:37,180 with the threefold axis. 193 00:12:37,180 --> 00:12:42,350 And then the two cubic arrangements 23, twofold axis 194 00:12:42,350 --> 00:12:45,980 out of the face normal to a cube, threefold axes out of 195 00:12:45,980 --> 00:12:50,990 all of the body diagonals, and 432. 196 00:12:50,990 --> 00:12:55,320 And what I'm going to do with you is to use these as 197 00:12:55,320 --> 00:13:02,610 worksheets to guide the logic of what we're doing, and we 198 00:13:02,610 --> 00:13:04,260 won't do every single one of them. 199 00:13:04,260 --> 00:13:07,970 I think once we do a few, the general principle is clear and 200 00:13:07,970 --> 00:13:09,990 the results there are before you. 201 00:13:09,990 --> 00:13:15,940 But what we'll take, then, is the axes n, the axes n22, and 202 00:13:15,940 --> 00:13:20,410 then the two cubic symmetries 23 and 432. 203 00:13:20,410 --> 00:13:25,080 Then, finally, we found 4-bar as a different rotation 204 00:13:25,080 --> 00:13:28,650 axis-like symmetry element, so I'll add that to the list. 205 00:13:31,530 --> 00:13:35,570 Then consider what we can add to these symmetries as 206 00:13:35,570 --> 00:13:43,150 extenders, and these are an additional element that we can 207 00:13:43,150 --> 00:13:44,590 add to what is already a 208 00:13:44,590 --> 00:13:47,080 self-contained nice little group. 209 00:13:47,080 --> 00:13:51,070 We muck things up by introducing another operation, 210 00:13:51,070 --> 00:13:54,000 and then we'll have to take combinations of that operation 211 00:13:54,000 --> 00:13:56,720 with all the symmetry elements that are there in the parent 212 00:13:56,720 --> 00:14:00,780 subgroup and see what new symmetry elements arise. 213 00:14:00,780 --> 00:14:04,860 The ground rules in these additions are fairly simple 214 00:14:04,860 --> 00:14:07,930 but nevertheless profound. 215 00:14:07,930 --> 00:14:21,730 The addition of the extender cannot create any new symmetry 216 00:14:21,730 --> 00:14:45,140 axes by its operation, and the reason for that should be 217 00:14:45,140 --> 00:14:47,150 quite clear. 218 00:14:47,150 --> 00:14:52,730 We obtained these combinations of rotation axes using Euler's 219 00:14:52,730 --> 00:14:59,110 principle, and that was thorough and orderly, 220 00:14:59,110 --> 00:15:01,140 systematic and complete. 221 00:15:01,140 --> 00:15:02,810 These are the only arrangements of 222 00:15:02,810 --> 00:15:06,550 crystallographic rotation axes that are possible. 223 00:15:06,550 --> 00:15:10,940 So if we add an extender that creates, for example, a new 224 00:15:10,940 --> 00:15:15,120 twofold axis in n22, it's either going to be something 225 00:15:15,120 --> 00:15:18,270 that we already have in this list, and therefore not 226 00:15:18,270 --> 00:15:20,710 interesting, or something that just cannot 227 00:15:20,710 --> 00:15:22,140 constitute a group. 228 00:15:22,140 --> 00:15:25,350 We'll take combinations of operations and we will never 229 00:15:25,350 --> 00:15:29,450 find a closed finite set. 230 00:15:29,450 --> 00:15:35,180 So if we add a mirror operation, a reflection 231 00:15:35,180 --> 00:15:41,960 operation as an extender, the addition of the reflection 232 00:15:41,960 --> 00:15:48,430 operation sigma must be either perpendicular to the axis. 233 00:15:48,430 --> 00:15:51,040 And what that's going to do is just flip the top part of the 234 00:15:51,040 --> 00:15:53,840 rotation axis and reflect it down to the bottom axis. 235 00:15:53,840 --> 00:15:56,080 Nothing new was created. 236 00:15:56,080 --> 00:16:00,200 And it could alternatively be parallel to the axis and 237 00:16:00,200 --> 00:16:01,700 passing through it. 238 00:16:01,700 --> 00:16:05,200 In that case, we've already seen the consequences of this 239 00:16:05,200 --> 00:16:09,440 addition in deriving the two-dimensional point groups, 240 00:16:09,440 --> 00:16:12,540 namely to 2mm, 3m, 4mm, and 6mm. 241 00:16:15,480 --> 00:16:16,740 No new axis is created. 242 00:16:19,390 --> 00:16:23,720 One of the additions that is something we mentioned last 243 00:16:23,720 --> 00:16:27,750 time is that in the case of the dihedral groups where we 244 00:16:27,750 --> 00:16:31,130 have an n-fold axis and then twofold axes 245 00:16:31,130 --> 00:16:33,400 arranged in some fashion-- 246 00:16:33,400 --> 00:16:35,050 not in some arbitrary fashion. 247 00:16:35,050 --> 00:16:39,740 It's going to be 2 pi over n times 1/2, depending on the 248 00:16:39,740 --> 00:16:41,380 n-fold axis-- 249 00:16:41,380 --> 00:16:45,460 we could pass the mirror operation 250 00:16:45,460 --> 00:16:47,040 through the twofold axis. 251 00:16:47,040 --> 00:16:49,570 And that's what we'll call a vertical mirror plane. 252 00:16:49,570 --> 00:16:52,990 But when there's more than one axis present, we could also 253 00:16:52,990 --> 00:16:59,450 put the reflection operation diagonally in between the 254 00:16:59,450 --> 00:17:01,560 twofold axes. 255 00:17:01,560 --> 00:17:06,609 So we're going to refer to this as a diagonal mirror. 256 00:17:06,609 --> 00:17:11,079 It's snaked in between the axes that are present. 257 00:17:11,079 --> 00:17:14,849 So, perpendicular reflection operation as an extender, one 258 00:17:14,849 --> 00:17:17,970 parallel to the axis and passing through it, or one 259 00:17:17,970 --> 00:17:23,220 that is diagonal passing in between them. 260 00:17:23,220 --> 00:17:26,829 And finally, that exhausts what we can do with 261 00:17:26,829 --> 00:17:29,850 reflection, but we could add inversion. 262 00:17:29,850 --> 00:17:34,340 And if we're not going create new axes, and we're going to 263 00:17:34,340 --> 00:17:36,730 get a point group-- a finite set of objects-- 264 00:17:36,730 --> 00:17:46,740 the inversion center has to be on one of the single axis, 265 00:17:46,740 --> 00:17:50,260 namely these groups n, or at the point of intersection, if 266 00:17:50,260 --> 00:17:51,510 there's more than one. 267 00:18:02,910 --> 00:18:05,700 So there's the job laid out for us. 268 00:18:05,700 --> 00:18:10,650 And we've got two theorems to aid us in quickly finding the 269 00:18:10,650 --> 00:18:13,030 new operations that arise-- 270 00:18:13,030 --> 00:18:15,870 in effect, taking a shortcut to establish the group 271 00:18:15,870 --> 00:18:20,050 multiplication table, and the two theorems are as follows. 272 00:18:20,050 --> 00:18:27,390 We saw that if you take a rotation operation A pi, that 273 00:18:27,390 --> 00:18:31,030 takes a first object that's right handed and rotates it to 274 00:18:31,030 --> 00:18:34,690 a second object that is also right handed. 275 00:18:34,690 --> 00:18:43,070 Then follow that by reflection in a mirror plane that is 276 00:18:43,070 --> 00:18:46,030 perpendicular to the axis. 277 00:18:46,030 --> 00:18:50,990 The net effect of going from 1 to number 3, which is left 278 00:18:50,990 --> 00:18:54,870 handed, is to create an inversion center at the point 279 00:18:54,870 --> 00:18:56,120 of intersection. 280 00:19:01,820 --> 00:19:03,880 So there's a theorem that we can write down once and for 281 00:19:03,880 --> 00:19:09,870 all and say that A pi followed by reflection in a mirror 282 00:19:09,870 --> 00:19:14,480 plane that's perpendicular to axis A has, as a net effect, 283 00:19:14,480 --> 00:19:15,800 the operation of inversion. 284 00:19:21,070 --> 00:19:23,620 And if this is not a twofold axis, it's something like a 285 00:19:23,620 --> 00:19:26,480 fourfold axis or a sixfold axis, that contains the 286 00:19:26,480 --> 00:19:30,190 operation A pi as part of the operations contained in that 287 00:19:30,190 --> 00:19:32,310 axis, and that's going to create an 288 00:19:32,310 --> 00:19:34,430 inversion center as well. 289 00:19:34,430 --> 00:19:39,470 So for any evenfold axis, add a perpendicular mirror plane, 290 00:19:39,470 --> 00:19:42,670 and automatically the inversion center comes up. 291 00:19:42,670 --> 00:19:46,180 And we can permute the order of these operations if we 292 00:19:46,180 --> 00:19:52,540 rotate by 180 degrees and then invert the net effect is 293 00:19:52,540 --> 00:19:55,940 reflection in the perpendicular mirror plane. 294 00:19:58,460 --> 00:20:01,570 So this is something that you permute around into three 295 00:20:01,570 --> 00:20:02,820 different combinations. 296 00:20:07,210 --> 00:20:11,470 Does the order of the operation make a difference? 297 00:20:11,470 --> 00:20:14,815 And the answer is no, the order is not important. 298 00:20:23,560 --> 00:20:28,290 For example, rotating and then reflecting is the same as 299 00:20:28,290 --> 00:20:30,180 reflecting and then rotating it. 300 00:20:30,180 --> 00:20:33,880 It went up at the same point, number 3. 301 00:20:33,880 --> 00:20:37,190 Remember a fancy word for this when we define what's meant by 302 00:20:37,190 --> 00:20:40,640 a group, that these groups are going to be what's called 303 00:20:40,640 --> 00:20:49,160 Abelian, and an Abelian group is where any combination of 304 00:20:49,160 --> 00:20:56,090 operations a followed by b is identical to b followed by a. 305 00:20:56,090 --> 00:21:01,190 I told you that great joke among mathematicians, what is 306 00:21:01,190 --> 00:21:03,920 purple and commutes? 307 00:21:03,920 --> 00:21:06,380 And the answer is an Abelian grape. 308 00:21:06,380 --> 00:21:07,400 Ha, ha, ha-- 309 00:21:07,400 --> 00:21:10,600 I think that's stupid, but it drives mathematicians into 310 00:21:10,600 --> 00:21:12,050 guffaws of laughter. 311 00:21:16,520 --> 00:21:22,530 A second theorem, and this is one that we've already seen in 312 00:21:22,530 --> 00:21:27,720 two dimensions, is that if you take an operation A alpha, 313 00:21:27,720 --> 00:21:34,780 pass a reflection operation sigma 1 through it, that to 314 00:21:34,780 --> 00:21:40,140 the effect of that is going to be another mirror plane that 315 00:21:40,140 --> 00:21:45,490 is alpha over 2 away from the first. 316 00:21:45,490 --> 00:21:48,080 Notice I'm a little bit sloppy in talking about reflection 317 00:21:48,080 --> 00:21:50,590 operation and mirror plane because if a reflection 318 00:21:50,590 --> 00:21:53,700 operation is there, that's all I need to have to say that 319 00:21:53,700 --> 00:21:56,440 this is the locus of a symmetry 320 00:21:56,440 --> 00:21:59,450 point, a mirror plane. 321 00:21:59,450 --> 00:22:01,280 So this is another one. 322 00:22:01,280 --> 00:22:05,080 Do you think that this an Abelian combination to say 323 00:22:05,080 --> 00:22:09,840 that A alpha followed by sigma 1 is equal to sigma 2. 324 00:22:14,240 --> 00:22:15,490 Is this Abelian? 325 00:22:17,886 --> 00:22:18,858 AUDIENCE: No. 326 00:22:18,858 --> 00:22:20,480 PROFESSOR: No, it's not. 327 00:22:20,480 --> 00:22:21,630 And how do you answer that? 328 00:22:21,630 --> 00:22:24,660 Not through any means more profound than just drawing it 329 00:22:24,660 --> 00:22:26,200 out and seeing what you get. 330 00:22:26,200 --> 00:22:29,490 Let's take 3n as an example. 331 00:22:29,490 --> 00:22:31,050 So let's do the operation. 332 00:22:31,050 --> 00:22:36,240 In this order, do the rotation A 2 pi over 3 to go from this 333 00:22:36,240 --> 00:22:41,200 one up to this one, and then let's reflect across this 334 00:22:41,200 --> 00:22:42,460 horizontal mirror plane. 335 00:22:42,460 --> 00:22:46,930 So this is 1 right handed, this is 2 right handed 336 00:22:46,930 --> 00:22:50,040 reflect, here's number three and it's left handed. 337 00:22:50,040 --> 00:22:54,990 So that's A 2 pi over 3 followed by 338 00:22:54,990 --> 00:22:59,420 reflection in sigma 1. 339 00:22:59,420 --> 00:23:04,060 And if we do the operations in reverse order, reflect in 340 00:23:04,060 --> 00:23:10,410 sigma 1 to go up to here, so this would be 2 prime and then 341 00:23:10,410 --> 00:23:14,070 rotate by 120 degrees. 342 00:23:14,070 --> 00:23:20,370 We would go to here to here, and that is not the same 343 00:23:20,370 --> 00:23:22,730 location as this one. 344 00:23:22,730 --> 00:23:26,920 So this is not equal to sigma 1 followed by A 2 pi over 3. 345 00:23:30,420 --> 00:23:34,530 How can you tell when is the combination of operations is 346 00:23:34,530 --> 00:23:36,930 going to be Abelian and when is it not? 347 00:23:36,930 --> 00:23:40,480 It sounds rather clumsy to state it in words, but 348 00:23:40,480 --> 00:23:44,910 whenever the two symmetry operations leave each other 349 00:23:44,910 --> 00:23:48,550 unmoved, then you can permute the order. 350 00:23:48,550 --> 00:23:52,330 For example, in this combination here the rotation 351 00:23:52,330 --> 00:23:53,730 leaves the mirror plane unchanged. 352 00:23:53,730 --> 00:23:56,150 It just spins around in its own plane. 353 00:23:56,150 --> 00:23:58,970 The mirror plane leaves the rotation unchanged, it just 354 00:23:58,970 --> 00:24:01,110 reflects it end to end. 355 00:24:01,110 --> 00:24:04,510 Here this rotation axis and these mirror planes obviously 356 00:24:04,510 --> 00:24:06,680 do not leave each other unchanged because the 357 00:24:06,680 --> 00:24:10,160 threefold axis rotates the mirror plane to 358 00:24:10,160 --> 00:24:12,180 two other new locations. 359 00:24:12,180 --> 00:24:15,100 So whenever that's the case, if the operations do not leave 360 00:24:15,100 --> 00:24:18,870 each other unchanged, the order makes a difference. 361 00:24:18,870 --> 00:24:23,560 So that's a useful thing to keep in mind. 362 00:24:23,560 --> 00:24:28,730 Let's now turn to these sheets that I passed out, and you 363 00:24:28,730 --> 00:24:32,020 might want to unstaple them because they're designed to go 364 00:24:32,020 --> 00:24:37,310 side by side, and there are three pages. 365 00:24:37,310 --> 00:24:48,360 What I've done across the top of the pages is to give the 366 00:24:48,360 --> 00:24:51,880 different combinations of crystallographic rotation axes 367 00:24:51,880 --> 00:24:53,560 that are possible. 368 00:24:53,560 --> 00:24:56,810 So going from the first sheet across to the last, there's 369 00:24:56,810 --> 00:24:59,200 the rotation axes by themselves-- 370 00:24:59,200 --> 00:25:01,940 one, two, three, four, and six. 371 00:25:01,940 --> 00:25:07,390 Then there are the groups of the form n22, 222, 32, 422, 372 00:25:07,390 --> 00:25:12,670 622, and then the two cubic combinations 23 and 432. 373 00:25:12,670 --> 00:25:14,960 So there are the 11 possible combination of 374 00:25:14,960 --> 00:25:19,110 crystallographic rotation axes, and stuck off by itself 375 00:25:19,110 --> 00:25:21,040 at the end is the oddball 4-bar. 376 00:25:30,390 --> 00:25:37,400 For the axes by themselves, the diagonal mirror addition 377 00:25:37,400 --> 00:25:38,530 is not defined. 378 00:25:38,530 --> 00:25:41,710 The diagonal mirror position by definition snakes in 379 00:25:41,710 --> 00:25:45,150 between axes that are there in the axial combination. 380 00:25:45,150 --> 00:25:48,500 If there's only one axis, that's not defined. 381 00:25:48,500 --> 00:25:52,250 But you could add a horizontal mirror plane. 382 00:25:52,250 --> 00:25:55,010 A horizontal mirror planes adds a onefold axis. 383 00:25:55,010 --> 00:25:57,330 It's just a mirror plane. 384 00:25:57,330 --> 00:25:59,490 So that's called m in the international 385 00:25:59,490 --> 00:26:01,500 notation for mirror. 386 00:26:01,500 --> 00:26:06,650 And it's called C sub S, C because it's a cyclic group, 387 00:26:06,650 --> 00:26:11,720 and the S stands for Spiegel. 388 00:26:11,720 --> 00:26:15,150 That's the Schoenflies notation. 389 00:26:15,150 --> 00:26:16,400 If we add a-- 390 00:26:18,830 --> 00:26:22,570 take two and add a horizontal mirror plane-- 391 00:26:22,570 --> 00:26:25,530 this is the thing that we sketched out here-- 392 00:26:25,530 --> 00:26:29,430 this gives a pattern of four objects, two that are up above 393 00:26:29,430 --> 00:26:32,740 the mirror plane of one chirality, two down below the 394 00:26:32,740 --> 00:26:36,150 mirror plane of opposite chirality. 395 00:26:36,150 --> 00:26:39,850 These projections of the patterns are down along the 396 00:26:39,850 --> 00:26:43,620 rotation axis, and when you see a dot in a circle, the 397 00:26:43,620 --> 00:26:48,010 circle represents the point that is down and the dot is 398 00:26:48,010 --> 00:26:49,500 one that's up on top. 399 00:26:49,500 --> 00:26:52,200 So there are two pairs of objects-- 400 00:26:52,200 --> 00:26:56,190 two right handed ones that are up and two left-handed ones 401 00:26:56,190 --> 00:26:57,790 that are down. 402 00:26:57,790 --> 00:27:02,800 And that, in international notation, is 2/m the twofold 403 00:27:02,800 --> 00:27:04,930 axis is over a mirror plane. 404 00:27:04,930 --> 00:27:07,840 So that's the way to make sense of the symbol and 405 00:27:07,840 --> 00:27:09,500 remember what it means. 406 00:27:09,500 --> 00:27:12,580 In the Schoenflies notation at C2. 407 00:27:12,580 --> 00:27:16,220 You've added a horizontal mirror plane as an extender. 408 00:27:20,950 --> 00:27:23,970 Going down the remaining ones in that list because they're 409 00:27:23,970 --> 00:27:25,420 all very, very similar-- 410 00:27:25,420 --> 00:27:28,990 add a mirror plane to a threefold axis, the triangle 411 00:27:28,990 --> 00:27:33,410 is reflected down to a triangle of motifs of opposite 412 00:27:33,410 --> 00:27:36,310 chirality, so that's 3 over m. 413 00:27:36,310 --> 00:27:40,552 For sure, it's called 6-bar, but let's forget about that. 414 00:27:40,552 --> 00:27:45,280 The Schoenflies notation C3H, and similarly there's a 4 over 415 00:27:45,280 --> 00:27:48,970 m, a square above, and a square of opposite handedness 416 00:27:48,970 --> 00:27:51,770 below the mirror plane. 417 00:27:51,770 --> 00:27:56,700 The horizontal mirror plane in all of these cases is shown as 418 00:27:56,700 --> 00:27:58,490 a bold line. 419 00:27:58,490 --> 00:28:02,060 They've added fainter vertical lines to give 420 00:28:02,060 --> 00:28:04,040 you an angular reference. 421 00:28:04,040 --> 00:28:07,560 Don't be confused by that in the Xerox copy-- 422 00:28:07,560 --> 00:28:10,290 the weight of the lines is not distinguished. 423 00:28:10,290 --> 00:28:13,460 So the only symmetry element in 4 over m, for example, 424 00:28:13,460 --> 00:28:17,670 should be shown by this solid bold circle that is in the 425 00:28:17,670 --> 00:28:18,920 plane of the paper. 426 00:28:18,920 --> 00:28:23,250 The two crossed lines are lighter and those are just as 427 00:28:23,250 --> 00:28:25,180 mutual orientations. 428 00:28:25,180 --> 00:28:27,430 And finally, there's another one of this form 429 00:28:27,430 --> 00:28:31,485 which is 6 over m. 430 00:28:31,485 --> 00:28:34,640 On all of the evenfold additions-- that is, to say 431 00:28:34,640 --> 00:28:36,320 everything but 3m-- 432 00:28:36,320 --> 00:28:42,040 if you add that horizontal mirror plane, it's going to be 433 00:28:42,040 --> 00:28:44,910 perpendicular to an operation A pi. 434 00:28:44,910 --> 00:28:48,870 Therefore, inversion pops up as one of the operations in 435 00:28:48,870 --> 00:28:54,340 the group, and 2 over m, 4 over m, 6 over m, those 436 00:28:54,340 --> 00:28:57,550 regular polyhedra of objects-- 437 00:28:57,550 --> 00:29:00,380 can be inverted through the point of intersection of the 438 00:29:00,380 --> 00:29:05,340 axis in the mirror plane, and that leaves the set unchanged. 439 00:29:05,340 --> 00:29:08,770 So if we go down to the bottom of the list and say, can we 440 00:29:08,770 --> 00:29:13,320 add inversion to these axial arrangements, in the case of 441 00:29:13,320 --> 00:29:19,120 2, 4, and 6-- 442 00:29:19,120 --> 00:29:22,550 no, adding inversion doesn't give you anything new, because 443 00:29:22,550 --> 00:29:30,490 that's already there in the groups of the form CNH, or 444 00:29:30,490 --> 00:29:33,760 2/m, 4/m, 6/m. 445 00:29:33,760 --> 00:29:38,410 Diagonal is not defined, so the only additional one that 446 00:29:38,410 --> 00:29:42,300 we could pick up by adding inversion is adding inversion 447 00:29:42,300 --> 00:29:44,930 to a threefold axis, and that's one we 448 00:29:44,930 --> 00:29:45,700 did the other day. 449 00:29:45,700 --> 00:29:49,010 Adding inversion to a threefold axis gives us three 450 00:29:49,010 --> 00:29:52,480 that are up of one handedness, three that are down of 451 00:29:52,480 --> 00:29:54,690 opposite handedness, and the orientation of 452 00:29:54,690 --> 00:29:57,220 the triangle is skewed. 453 00:29:57,220 --> 00:29:59,460 That is called 3-bar. 454 00:29:59,460 --> 00:30:02,270 But it's just nothing more than a threefold axis with an 455 00:30:02,270 --> 00:30:04,300 inversion center sitting on it. 456 00:30:04,300 --> 00:30:05,815 Schoenflies' notation is C3i. 457 00:30:10,210 --> 00:30:13,750 Then the only other additions to the single axes are adding 458 00:30:13,750 --> 00:30:18,360 the mirror plane in a vertical fashion passing through the 459 00:30:18,360 --> 00:30:21,720 axis, and these we've already seen in two dimensions so we 460 00:30:21,720 --> 00:30:23,870 don't have to spend much time on them. 461 00:30:23,870 --> 00:30:27,910 There's 2 mm, which is C2V. 462 00:30:27,910 --> 00:30:29,210 3m-- 463 00:30:29,210 --> 00:30:32,280 all the mirror planes result upon adding a single mirror 464 00:30:32,280 --> 00:30:33,960 plane to the threefold axis-- 465 00:30:33,960 --> 00:30:36,330 4 mm, and 6 mm. 466 00:30:36,330 --> 00:30:38,650 These are just three-dimensional extensions 467 00:30:38,650 --> 00:30:41,700 parallel to the axis of the symmetry that we already 468 00:30:41,700 --> 00:30:45,110 derived for a plane in space in the 469 00:30:45,110 --> 00:30:50,290 two-dimensional point groups. 470 00:30:50,290 --> 00:30:51,650 So we're almost half done. 471 00:30:51,650 --> 00:30:52,850 It's easy, isn't it? 472 00:30:52,850 --> 00:30:54,100 Any questions at this point? 473 00:31:01,640 --> 00:31:04,000 So the only ones that are really new that we haven't 474 00:31:04,000 --> 00:31:08,600 seen already in two dimensions are the rotation axis 475 00:31:08,600 --> 00:31:10,170 perpendicular to the mirrored plane. 476 00:31:12,840 --> 00:31:16,060 That brings us to roughly the middle of the second sheet, 477 00:31:16,060 --> 00:31:22,180 and the arrangement of axes 222 represents three mutually 478 00:31:22,180 --> 00:31:26,020 orthogonal twofold axes. 479 00:31:26,020 --> 00:31:33,050 If we add a horizontal mirror plane, that's going to put an 480 00:31:33,050 --> 00:31:36,070 inversion center at the point of intersection because that 481 00:31:36,070 --> 00:31:39,220 horizontal reflection is going to be sitting normal to a 482 00:31:39,220 --> 00:31:41,620 twofold axis. 483 00:31:41,620 --> 00:31:44,845 And then since the other twofold axes see an inversion 484 00:31:44,845 --> 00:31:48,250 center sitting on it, there's a mirror plane perpendicular 485 00:31:48,250 --> 00:31:50,600 to those other twofold axes. 486 00:31:50,600 --> 00:31:54,020 So adding again to go through that again, adding inversion 487 00:31:54,020 --> 00:31:58,170 to the point of intersection of the twofold axes creates a 488 00:31:58,170 --> 00:32:01,820 mirror plane perpendicular to each of those twofold axes. 489 00:32:01,820 --> 00:32:05,900 So it becomes 2/m 2/m 2/m-- 490 00:32:05,900 --> 00:32:09,910 each of the twofold axes acquires a different sort of 491 00:32:09,910 --> 00:32:13,740 mirror plane perpendicular to its orientation. 492 00:32:13,740 --> 00:32:17,310 And 2/m 2/m 2/m, even though I love saying it, 493 00:32:17,310 --> 00:32:18,960 is kind of a mouthful. 494 00:32:18,960 --> 00:32:23,470 So that's very often abbreviated to just mmm, which 495 00:32:23,470 --> 00:32:26,430 really is a nice exclamation when you see what a lovely 496 00:32:26,430 --> 00:32:29,510 symmetry it is. 497 00:32:29,510 --> 00:32:35,620 The pattern shown in the diagram to the left is just 498 00:32:35,620 --> 00:32:39,060 the pattern of 222 with the objects reflected up or 499 00:32:39,060 --> 00:32:43,700 reflected down, and you get a total of 8. 500 00:32:43,700 --> 00:32:46,240 That means that there are eight operations in the group, 501 00:32:46,240 --> 00:32:49,380 and you can add up quickly what they are. 502 00:32:49,380 --> 00:32:54,250 They are the three operations of rotating by pi, the 503 00:32:54,250 --> 00:32:59,300 operation of identity, then three mirror planes because 504 00:32:59,300 --> 00:33:01,150 the mirror planes all indistinct, and then the 505 00:33:01,150 --> 00:33:02,240 operation of inversion. 506 00:33:02,240 --> 00:33:05,390 So there are the eight operations that are present 507 00:33:05,390 --> 00:33:08,300 that generate the objects from a single one. 508 00:33:11,630 --> 00:33:20,590 Add a horizontal mirror plane to 32 and you get a threefold 509 00:33:20,590 --> 00:33:23,330 axis perpendicular to the threefold axis, so you write 510 00:33:23,330 --> 00:33:26,200 that as 3m. 511 00:33:26,200 --> 00:33:29,520 That mirror plane now passes through the horizontal twofold 512 00:33:29,520 --> 00:33:33,470 axes, and a twofold axis with a mirror plane passing through 513 00:33:33,470 --> 00:33:37,330 it wants another mirror plane 90 degrees away, so a new 514 00:33:37,330 --> 00:33:41,060 mirror plane pops up passing through each of the twofold 515 00:33:41,060 --> 00:33:46,150 axes and perpendicular to the horizontal mirror plane. 516 00:33:46,150 --> 00:33:49,150 Those mirror planes are in the same direction 517 00:33:49,150 --> 00:33:50,300 as the twofold axis. 518 00:33:50,300 --> 00:33:55,610 They're not perpendicular to it, and when symmetry elements 519 00:33:55,610 --> 00:33:58,400 are parallel to a common direction you write them out 520 00:33:58,400 --> 00:34:01,670 on the same line, so here's a case of a mixed metaphor. 521 00:34:01,670 --> 00:34:04,350 One mirror plane that's perpendicular to an axis, the 522 00:34:04,350 --> 00:34:07,460 threefold axis, so you write that as 3 over m. 523 00:34:07,460 --> 00:34:10,320 The other twofold axes have mirror points passing through 524 00:34:10,320 --> 00:34:14,699 them, so you write that as 2m and the threefold axes makes 525 00:34:14,699 --> 00:34:20,380 all of those 2m symmetries equivalent to one another. 526 00:34:20,380 --> 00:34:25,489 Pattern, and this is true for all of these symmetries, is 527 00:34:25,489 --> 00:34:29,790 nothing more than the pattern of the subgroup repeated by 528 00:34:29,790 --> 00:34:33,520 the one operation that you've added as an extender . 529 00:34:33,520 --> 00:34:37,949 So if you look at the pattern of 32, object all of the same 530 00:34:37,949 --> 00:34:40,710 chirality, all apparently up and down, and add the 531 00:34:40,710 --> 00:34:42,139 horizontal mirror plane. 532 00:34:42,139 --> 00:34:45,330 The one that is up goes down, the one that's down goes up, 533 00:34:45,330 --> 00:34:47,550 and you get a set of four around each 534 00:34:47,550 --> 00:34:48,800 of the twofold axes. 535 00:34:51,770 --> 00:34:55,989 422 behaves very similarly. 536 00:34:55,989 --> 00:34:59,500 The pattern is just the pattern of 422 reflected in a 537 00:34:59,500 --> 00:35:08,040 plane, so you have a set of 16 objects-- 538 00:35:08,040 --> 00:35:10,910 eight of them up of one chirality, those are the solid 539 00:35:10,910 --> 00:35:15,250 dots, if you will, and then another two, four, six, eight 540 00:35:15,250 --> 00:35:18,460 that are down of opposite chirality. 541 00:35:18,460 --> 00:35:22,190 You have a mirror plane that you added as the horizontal 542 00:35:22,190 --> 00:35:25,070 mirror plane perpendicular to the fourfold axis. 543 00:35:25,070 --> 00:35:27,880 All of the horizontal twofold axes see a mirror plane 544 00:35:27,880 --> 00:35:28,700 passing through them. 545 00:35:28,700 --> 00:35:31,300 So they've got to have a mirror plane that's 90 degrees 546 00:35:31,300 --> 00:35:33,510 away, and those are the vertical mirror planes. 547 00:35:36,250 --> 00:35:38,640 Either the point is obvious now or you're not following me 548 00:35:38,640 --> 00:35:41,770 at all, so I'll simply say on the left-hand column of the 549 00:35:41,770 --> 00:35:46,000 final sheet, 622 with a horizontal mirror plane goes 550 00:35:46,000 --> 00:35:51,050 to 6/m, 2/m, 2/m, called D6h in Schoenflies. 551 00:35:55,590 --> 00:35:57,700 I'll leave the cubic ones until last. 552 00:35:57,700 --> 00:36:01,250 They're not nearly as bad as they seem. 553 00:36:01,250 --> 00:36:03,760 But obviously they have a lot of symmetry all over the 554 00:36:03,760 --> 00:36:08,285 place, and we'll want to take a closer look at the patterns. 555 00:36:14,830 --> 00:36:21,970 The next extender is a vertical mirror plane, and 556 00:36:21,970 --> 00:36:27,410 we've already got the groups of the form cnv, because they 557 00:36:27,410 --> 00:36:30,860 are just extensions in a third dimension of one set we've 558 00:36:30,860 --> 00:36:33,705 seen in two dimensions. 559 00:36:33,705 --> 00:36:39,610 If we try adding the vertical mirror plane, which is defined 560 00:36:39,610 --> 00:36:42,560 as passing through the principal axis of high 561 00:36:42,560 --> 00:36:47,740 symmetry and perpendicular to the twofold axes, we've 562 00:36:47,740 --> 00:36:54,990 already encountered those in every group except 32. 563 00:36:54,990 --> 00:37:00,300 2/m, 2/m, 2/m already has the vertical mirror plane. 564 00:37:00,300 --> 00:37:05,570 The same is true of 3/mm2. 565 00:37:05,570 --> 00:37:10,940 The same is true of 4/m, 2/m, 2/m and 6/m, 2/m, 2/m. 566 00:37:10,940 --> 00:37:13,600 The vertical mirror plane comes in automatically when we 567 00:37:13,600 --> 00:37:16,550 add the horizontal mirror plane as the extender, So 568 00:37:16,550 --> 00:37:20,270 nothing new there at all. 569 00:37:20,270 --> 00:37:24,780 And now we come to one that is interesting. 570 00:37:24,780 --> 00:37:29,210 This is the diagonal mirror plane. 571 00:37:29,210 --> 00:37:33,660 And I'll do this one slowly and then in some detail 572 00:37:33,660 --> 00:37:37,090 because it's another example of how we would trip over 573 00:37:37,090 --> 00:37:40,600 something if we were not clever enough to think of it. 574 00:37:44,220 --> 00:37:56,280 Let me begin by drawing the three orthogonal axes of 222. 575 00:37:56,280 --> 00:38:00,660 So we have one object that's up and one object that's down. 576 00:38:00,660 --> 00:38:04,550 The twofold axis perpendicular to the board will take this 577 00:38:04,550 --> 00:38:07,300 one that's down and rotate it to here and take this one 578 00:38:07,300 --> 00:38:09,630 that's up and rotate it to here. 579 00:38:09,630 --> 00:38:15,770 So that is the pattern of 222. 580 00:38:15,770 --> 00:38:25,360 We can snake a mirror plane in between the twofold axes, and 581 00:38:25,360 --> 00:38:29,090 a mirror plane passing through the vertical twofold axis has 582 00:38:29,090 --> 00:38:33,130 to be accompanied by one that's 90 degrees away, half 583 00:38:33,130 --> 00:38:37,000 the throw of the axis, which comes from our theorem that 584 00:38:37,000 --> 00:38:42,580 says A pi dot sigma vertical has to be equal to a sigma 585 00:38:42,580 --> 00:38:46,070 prime that's vertical and pi/2 away from the first. 586 00:38:48,890 --> 00:38:52,470 In terms of the pattern, this second mirror plane that comes 587 00:38:52,470 --> 00:38:56,500 up is going to reflect this one across to here, it's going 588 00:38:56,500 --> 00:39:00,680 to reflect this one across to here, it's going to take this 589 00:39:00,680 --> 00:39:03,730 one and reflect it over to here, and take this one and 590 00:39:03,730 --> 00:39:06,200 reflect it to the right, as well. 591 00:39:06,200 --> 00:39:11,590 So now we've got a total of eight objects. 592 00:39:11,590 --> 00:39:16,520 This one is right handed, and this one is also right handed 593 00:39:16,520 --> 00:39:18,660 because we repeated it by rotation. 594 00:39:18,660 --> 00:39:22,650 Then we reflected that pair, so these two are left handed, 595 00:39:22,650 --> 00:39:25,500 and reflect it across the other diagonal mirror plane. 596 00:39:25,500 --> 00:39:28,340 They're back to right handed again, and these have to be 597 00:39:28,340 --> 00:39:30,370 left handed as well, so there's the pattern. 598 00:39:36,280 --> 00:39:39,990 Is there an inversion center in this pattern? 599 00:39:39,990 --> 00:39:44,870 The answer is no, because there is no mirror plane that 600 00:39:44,870 --> 00:39:50,060 is perpendicular to a twofold axis, so this point group has 601 00:39:50,060 --> 00:39:51,560 no inversion in it. 602 00:39:51,560 --> 00:39:56,500 It's said to be acentric, without a inversion center. 603 00:39:56,500 --> 00:39:59,040 Do we know how everything is related to everything else? 604 00:39:59,040 --> 00:40:01,300 We know how this one is related to this 605 00:40:01,300 --> 00:40:03,460 one by twofold rotation. 606 00:40:03,460 --> 00:40:06,260 We know how this one is related to this one, and 607 00:40:06,260 --> 00:40:09,740 that's by reflection across the mirror plane. 608 00:40:09,740 --> 00:40:13,340 How is the first one related to this one? 609 00:40:17,400 --> 00:40:23,510 How do you get it through 90 degrees and then pop it down 610 00:40:23,510 --> 00:40:25,306 and change the chirality? 611 00:40:25,306 --> 00:40:26,614 AUDIENCE: [INAUDIBLE]? 612 00:40:26,614 --> 00:40:29,580 PROFESSOR: You can't do that in one step. 613 00:40:29,580 --> 00:40:31,720 You've got to do just what I said in words. 614 00:40:31,720 --> 00:40:35,020 Rotate it 90 degrees, don't put it down yet, and reflect 615 00:40:35,020 --> 00:40:37,430 it down in a horizontal mirror plane. 616 00:40:37,430 --> 00:40:40,170 This horizontal mirror plane is not a symmetry element, 617 00:40:40,170 --> 00:40:43,270 it's part of the operation that's necessary to get us 618 00:40:43,270 --> 00:40:48,050 from this guy over to this guy of opposite chirality. 619 00:40:48,050 --> 00:40:51,330 So this is an example a new type of 620 00:40:51,330 --> 00:40:53,970 operation, a 2 step operation. 621 00:40:53,970 --> 00:40:57,480 This is a 90-degree rotoreflection axis. 622 00:41:01,300 --> 00:41:04,940 And again, I feel compelled to put the axis in quotation 623 00:41:04,940 --> 00:41:10,090 marks, because really it's a pair of operations that leaves 624 00:41:10,090 --> 00:41:12,390 only a point unmoved, so it's really a 625 00:41:12,390 --> 00:41:13,640 point symmetry element. 626 00:41:24,850 --> 00:41:29,020 There's another way of describing the same thing, and 627 00:41:29,020 --> 00:41:31,220 that would be rotoinversion. 628 00:41:31,220 --> 00:41:37,720 We could rotate by 90 degrees in the reverse sense, not yet 629 00:41:37,720 --> 00:41:45,800 put it down, and invert, and we get, again, 630 00:41:45,800 --> 00:41:47,940 the relation between-- 631 00:41:47,940 --> 00:41:50,460 what did I do? 632 00:41:50,460 --> 00:41:51,940 Rotate it here, yes-- 633 00:41:51,940 --> 00:41:55,375 rotate to here and then invert, and we got that one. 634 00:41:58,050 --> 00:42:02,240 So that's another way of defining the relation. 635 00:42:02,240 --> 00:42:06,650 A 90-degree rotoreflection is a minus 90-degree 636 00:42:06,650 --> 00:42:07,900 rotoinversion. 637 00:42:16,030 --> 00:42:21,260 So you pays your money and you takes your choice, and what 638 00:42:21,260 --> 00:42:26,080 the groundbreakers who went before us did was to take 639 00:42:26,080 --> 00:42:27,390 rotoinversion. 640 00:42:27,390 --> 00:42:31,030 as the operation to survive. 641 00:42:31,030 --> 00:42:35,270 So here is, if we were not smart enough to invent it, our 642 00:42:35,270 --> 00:42:37,530 4-bar rotoinversion axis. 643 00:42:45,620 --> 00:42:48,100 So if we hadn't been clever enough-- and it would take a 644 00:42:48,100 --> 00:42:52,290 pretty clever, devious mind to invent a 2 step rotoinversion 645 00:42:52,290 --> 00:42:55,060 operation-- 646 00:42:55,060 --> 00:42:58,680 as soon as we added this sort of extender, a diagonal mirror 647 00:42:58,680 --> 00:43:03,540 plane to a group of the form n22, we would have found that 648 00:43:03,540 --> 00:43:06,600 there was a new sort of transformation that could not 649 00:43:06,600 --> 00:43:09,190 be described any more simply than to say take 650 00:43:09,190 --> 00:43:10,730 two steps to do it. 651 00:43:10,730 --> 00:43:14,750 Rotate 90 degrees and then invert. 652 00:43:14,750 --> 00:43:20,960 So we have to add the operation of A pi over 2-bar 653 00:43:20,960 --> 00:43:23,675 to our basic bag of tricks for generating patterns. 654 00:43:36,480 --> 00:43:43,790 The group 32 is also a group to which we can add a diagonal 655 00:43:43,790 --> 00:43:45,810 mirror plane. 656 00:43:45,810 --> 00:43:51,390 And if you do that, again, the bold lines that are mirror 657 00:43:51,390 --> 00:43:56,760 planes in that group and the axes that are faint lines are 658 00:43:56,760 --> 00:43:58,560 easy to mix up. 659 00:43:58,560 --> 00:44:15,700 But if we examine that group without the confusing lines, 660 00:44:15,700 --> 00:44:18,342 here are the twofold axes. 661 00:44:18,342 --> 00:44:22,780 And remember that they are all equivalent to one another by 662 00:44:22,780 --> 00:44:23,900 the threefold axis. 663 00:44:23,900 --> 00:44:27,700 So this axial group is the group 32. 664 00:44:27,700 --> 00:44:31,330 If we put down a mirror plane interleaved between the 665 00:44:31,330 --> 00:44:36,170 twofold axes, notice that each of those mirror is 666 00:44:36,170 --> 00:44:38,960 perpendicular to a twofold axis. 667 00:44:38,960 --> 00:44:43,040 So we can write this as 2/m. 668 00:44:43,040 --> 00:44:46,700 2/m means there's an inversion center at the intersection of 669 00:44:46,700 --> 00:44:49,430 the twofold axes with the mirror planes. 670 00:44:49,430 --> 00:44:54,260 And a inversion center sitting on a threefold axis makes it a 671 00:44:54,260 --> 00:45:00,100 3-bar axis, so this group is called 3-bar 2/m. 672 00:45:00,100 --> 00:45:05,600 And the pattern is what a twofold axis would do. 673 00:45:05,600 --> 00:45:07,990 These would be up-down-right ones. 674 00:45:07,990 --> 00:45:13,490 Reflect those and you get an up-down pair of left ones. 675 00:45:13,490 --> 00:45:16,870 Reflect again and you get an up-down pair of 676 00:45:16,870 --> 00:45:18,330 right-handed ones. 677 00:45:18,330 --> 00:45:24,720 Reflect yet again, and you get an up-down pair 678 00:45:24,720 --> 00:45:26,590 of left-handed ones. 679 00:45:26,590 --> 00:45:32,150 Reflect still once more, and you get a down-up pair of 680 00:45:32,150 --> 00:45:33,160 right-handed ones. 681 00:45:33,160 --> 00:45:38,030 One more time, up-down left-handed ones. 682 00:45:38,030 --> 00:45:43,410 So you get a total of 2, 4, 6, 8, 10, 12 points. 683 00:45:43,410 --> 00:45:49,010 So this is 3-bar 2/m in Schoenflies notation D3-- 684 00:45:49,010 --> 00:45:51,150 that's the symbol 32-- 685 00:45:51,150 --> 00:45:54,140 with a diagonal mirror plane added. 686 00:45:54,140 --> 00:45:56,475 You indicate that by a d in the subscript. 687 00:46:09,080 --> 00:46:09,400 OK. 688 00:46:09,400 --> 00:46:12,110 I think that's probably a point where you're more than 689 00:46:12,110 --> 00:46:13,650 ready for a break. 690 00:46:13,650 --> 00:46:15,705 We've got very few yet to do. 691 00:46:18,370 --> 00:46:22,450 One surprise is that if you add diagonal mirror planes to 692 00:46:22,450 --> 00:46:27,530 422 and 622, you get perfectly lovely exquisite groups but 693 00:46:27,530 --> 00:46:30,030 they're not crystallographic, so we don't 694 00:46:30,030 --> 00:46:31,530 include them in our list. 695 00:46:31,530 --> 00:46:34,330 But we'll discuss them when we come back, and then take a 696 00:46:34,330 --> 00:46:38,970 look also at the cubic symmetries. 697 00:46:38,970 --> 00:46:42,270 Impossible to draw, but really not all that difficult to 698 00:46:42,270 --> 00:46:44,800 understand. 699 00:46:44,800 --> 00:46:48,760 So let's stop at this point and take a stretch. 700 00:46:48,760 --> 00:46:50,080 People are stretching already. 701 00:46:50,080 --> 00:46:51,330 They need it. 702 00:46:53,300 --> 00:46:55,360 We'll resume in 10 minutes' time.