1 00:00:07,110 --> 00:00:09,890 PROFESSOR: --questions about what we've done. 2 00:00:09,890 --> 00:00:13,460 I think it's been fast, but hopefully if you understand 3 00:00:13,460 --> 00:00:16,500 the principles, we didn't go too fast. 4 00:00:16,500 --> 00:00:19,190 But any questions on what we've done? 5 00:00:24,470 --> 00:00:25,720 I guess you haven't had a chance to think 6 00:00:25,720 --> 00:00:27,980 of questions yet. 7 00:00:27,980 --> 00:00:33,240 Let me take care of our oddball symmetry at the end of 8 00:00:33,240 --> 00:00:38,150 the chart that I handed out-- and this 4 bar-- 9 00:00:38,150 --> 00:00:42,110 and ask what we can do there. 10 00:00:42,110 --> 00:00:50,060 Having discovered the four bar operation in 2, 2, 2 with 11 00:00:50,060 --> 00:00:53,960 diagonal mirror planes, we can consider that as a new type of 12 00:00:53,960 --> 00:00:55,650 symmetry element. 13 00:00:55,650 --> 00:01:02,450 And this is the symbol for it. 14 00:01:02,450 --> 00:01:07,430 And this would take a pair of objects and repeat them by a 15 00:01:07,430 --> 00:01:09,440 180 degree rotation. 16 00:01:09,440 --> 00:01:13,640 And then another pair of opposite handedness, opposite 17 00:01:13,640 --> 00:01:18,980 chirality would be rotated 90 degrees and inverted, rotated 18 00:01:18,980 --> 00:01:21,210 90 degrees and inverted. 19 00:01:21,210 --> 00:01:29,110 And I mentioned last time that a solid that has this shape is 20 00:01:29,110 --> 00:01:31,130 something called a sphenoid. 21 00:01:34,700 --> 00:01:39,580 And the 4 bar axis takes a pair of faces that are up and 22 00:01:39,580 --> 00:01:43,940 a pair of faces that are down and skewed by 90 degrees. 23 00:01:43,940 --> 00:01:45,890 Now what can we do? 24 00:01:45,890 --> 00:01:52,780 If we add a mirror plane that's perpendicular to the 4 25 00:01:52,780 --> 00:01:55,850 bar axis, the right-handed one goes down, the 26 00:01:55,850 --> 00:01:57,480 left-handed one comes up. 27 00:01:57,480 --> 00:02:03,200 And it becomes simply 4 over m, which we've already got. 28 00:02:03,200 --> 00:02:08,725 If we add a vertical mirror plane through the 4 bar, if 29 00:02:08,725 --> 00:02:18,040 you look a bit earlier at 4 bar 2m, now we've got the 4 30 00:02:18,040 --> 00:02:21,420 bar, now we've got the mirror plane. 31 00:02:21,420 --> 00:02:26,070 Not surprisingly, we get the two mirror planes. 32 00:02:26,070 --> 00:02:41,200 And so S4v is going to be the same as 4 bar 2m. 33 00:02:44,380 --> 00:02:48,400 And that's something we already have D2d. 34 00:02:48,400 --> 00:02:50,030 So that exhausts the possibilities. 35 00:02:50,030 --> 00:02:51,595 4 bar just stands by itself. 36 00:02:55,220 --> 00:02:58,790 At a vertical mirror plane, it's D2d 4 bar 2m. 37 00:02:58,790 --> 00:03:01,920 At a horizontal mirror plane, it becomes 4 over m. 38 00:03:01,920 --> 00:03:08,300 The diagonal mirror plane is not possible. 39 00:03:08,300 --> 00:03:11,500 There's nothing to place the mirror plane diagonal to. 40 00:03:11,500 --> 00:03:14,290 And if you add inversion, it changes into 4 over m. 41 00:03:14,290 --> 00:03:16,100 So this one is an odd group. 42 00:03:16,100 --> 00:03:19,330 It sits by itself. 43 00:03:19,330 --> 00:03:21,860 Except that we could start there, and add a vertical 44 00:03:21,860 --> 00:03:27,070 mirror plane, and get 4 bar 2m once more. 45 00:03:27,070 --> 00:03:31,160 That leaves the cubic ones, which are really easy to deal 46 00:03:31,160 --> 00:03:33,490 with because a cube has such high symmetry. 47 00:03:33,490 --> 00:03:35,450 We all know what cubes look like. 48 00:03:35,450 --> 00:03:39,210 But when you show arrangements of motifs, it gets a little 49 00:03:39,210 --> 00:03:41,070 bit confusing. 50 00:03:41,070 --> 00:03:44,295 So let's take a look at the tetrahedral groups. 51 00:03:47,630 --> 00:03:53,360 T is a combination of twofold axes coming out in directions 52 00:03:53,360 --> 00:03:55,670 corresponding to face normals to a cube. 53 00:04:00,590 --> 00:04:02,865 So these are the twofold axes. 54 00:04:08,510 --> 00:04:14,140 And then there are threefold axes coming out of directions 55 00:04:14,140 --> 00:04:17,760 that correspond to the face diagonal. 56 00:04:17,760 --> 00:04:22,060 So this is the jack-o'-lantern stereographic projection, 57 00:04:22,060 --> 00:04:27,410 which is something I love to draw at this time of year. 58 00:04:27,410 --> 00:04:29,360 What will the pattern look like? 59 00:04:29,360 --> 00:04:33,760 Well, it's going to look like 2, 2, 2. 60 00:04:37,010 --> 00:04:38,200 So that's a subgroup. 61 00:04:38,200 --> 00:04:42,050 So let's draw in a pair of objects on either side of the 62 00:04:42,050 --> 00:04:45,774 twofold axis, and that's the pattern of 2, 2, 2. 63 00:04:45,774 --> 00:04:50,420 Now this guy here is lurking off of a threefold axis. 64 00:04:50,420 --> 00:04:54,570 So that threefold axis is going to repeat it three 65 00:04:54,570 --> 00:04:58,800 locations that are 120 degrees apart. 66 00:04:58,800 --> 00:05:03,700 And so there's going to be a triangle of objects up above. 67 00:05:03,700 --> 00:05:10,660 This threefold axis is going to reproduce this into a 68 00:05:10,660 --> 00:05:13,660 triangle of objects that are down below. 69 00:05:13,660 --> 00:05:17,100 The twofold axis will take this triangle of objects and 70 00:05:17,100 --> 00:05:19,390 move it over to here. 71 00:05:19,390 --> 00:05:21,090 Also, up. 72 00:05:21,090 --> 00:05:24,110 And the twofold axis that's vertical will take this 73 00:05:24,110 --> 00:05:29,450 triangle and move it down to three that are below. 74 00:05:29,450 --> 00:05:31,540 So it looks very complicated in projection. 75 00:05:31,540 --> 00:05:35,700 But it's simply a planar triangle of atoms here rotated 76 00:05:35,700 --> 00:05:36,740 180 degrees. 77 00:05:36,740 --> 00:05:38,330 So you've got one like this, one like this. 78 00:05:38,330 --> 00:05:41,925 And then down below, two other planar triangles of objects. 79 00:05:47,700 --> 00:05:58,050 Remembering what the symmetry of a tetrahedron looks like, 80 00:05:58,050 --> 00:06:04,150 you can draw this arrangement of symmetry elements relative 81 00:06:04,150 --> 00:06:06,510 to a tetrahedron. 82 00:06:06,510 --> 00:06:09,890 And the threefold axes now are coming out 83 00:06:09,890 --> 00:06:11,365 normal to the faces. 84 00:06:11,365 --> 00:06:14,160 The twofold axes coming out normal to the edges. 85 00:06:16,810 --> 00:06:20,800 And imagine that we put a triangle on this face, a 86 00:06:20,800 --> 00:06:24,700 similar triangle on the face behind, and a triangle 87 00:06:24,700 --> 00:06:27,300 pointing in the other direction down this way. 88 00:06:27,300 --> 00:06:28,830 These are the three below. 89 00:06:28,830 --> 00:06:31,960 So that's the pattern of 2, 3. 90 00:06:31,960 --> 00:06:35,400 Take a tetrahedron, smack a triangle on the two upper 91 00:06:35,400 --> 00:06:39,130 faces, and an equilateral triangle on 92 00:06:39,130 --> 00:06:41,590 the two lower faces. 93 00:06:41,590 --> 00:06:44,880 And the threefold axis that comes out here comes out of 94 00:06:44,880 --> 00:06:47,000 the center of this triangle and out of the 95 00:06:47,000 --> 00:06:48,250 center of this triangle. 96 00:06:59,910 --> 00:07:06,230 The next one that you can get from this is Th, the 97 00:07:06,230 --> 00:07:08,530 tetrahedral arrangement of axes with a 98 00:07:08,530 --> 00:07:11,470 horizontal mirror plane. 99 00:07:11,470 --> 00:07:14,290 If you put in a horizontal mirror plane, it's going to 100 00:07:14,290 --> 00:07:19,850 take this triangle and reflect it down. 101 00:07:19,850 --> 00:07:22,540 It's going to take this triangle and reflect it up. 102 00:07:22,540 --> 00:07:26,336 And they will overlap in projection. 103 00:07:26,336 --> 00:07:30,440 And so that's the pattern for Th. 104 00:07:30,440 --> 00:07:32,470 There's a mirror plane perpendicular 105 00:07:32,470 --> 00:07:34,880 to a twofold axis. 106 00:07:34,880 --> 00:07:36,920 That creates an inversion center. 107 00:07:36,920 --> 00:07:39,200 So we can call this a 3 bar axis. 108 00:07:39,200 --> 00:07:44,360 So that's Th 2 over m 3 bar. 109 00:07:50,720 --> 00:07:56,715 So just have triangles on the upper and lower faces as well. 110 00:08:03,270 --> 00:08:11,970 The one remaining one is Td. 111 00:08:17,760 --> 00:08:19,850 Here are the twofold axes. 112 00:08:19,850 --> 00:08:24,620 Here are the threefold axes. 113 00:08:24,620 --> 00:08:28,780 And now if we put a diagonal mirror plane in, diagonal with 114 00:08:28,780 --> 00:08:29,690 respect to what? 115 00:08:29,690 --> 00:08:32,390 Well, diagonal with respect to these twofold axes. 116 00:08:32,390 --> 00:08:37,140 So this mirror plane goes down like this, passes through a 117 00:08:37,140 --> 00:08:38,110 twofold axis. 118 00:08:38,110 --> 00:08:42,799 There must be a mirror plane 90 degrees away. 119 00:08:42,799 --> 00:08:50,170 And the threefold axis is going to repeat these mirror 120 00:08:50,170 --> 00:08:58,270 planes so that we get mirror planes that are at an angle 121 00:08:58,270 --> 00:09:01,210 with respect to the vertical twofold axis. 122 00:09:01,210 --> 00:09:05,100 And this is Td. 123 00:09:05,100 --> 00:09:10,800 And this, in the international notation, is called-- if we 124 00:09:10,800 --> 00:09:12,320 can figure it out-- 125 00:09:12,320 --> 00:09:16,470 the diagonal mirror planes with respect to these twofold 126 00:09:16,470 --> 00:09:20,250 axes have changed them into 4 bar axes. 127 00:09:20,250 --> 00:09:26,740 So this is called 4 bar 3m, which doesn't 128 00:09:26,740 --> 00:09:28,230 look cubic at all. 129 00:09:28,230 --> 00:09:29,555 So that's a little bit deceptive. 130 00:09:36,970 --> 00:09:41,960 I am foolhardy for even trying to do this. 131 00:09:41,960 --> 00:09:44,280 And so I don't think I'm going to do it. 132 00:09:44,280 --> 00:09:50,930 4, 3, 2, we know what that looks like. 133 00:09:50,930 --> 00:09:53,930 That is fourfold axes coming out in directions that 134 00:09:53,930 --> 00:10:01,115 correspond to the face normals to a cube. 135 00:10:01,115 --> 00:10:06,110 A threefold axis coming out of body diagonals. 136 00:10:06,110 --> 00:10:09,460 Twofold axes between all of the fourfold axes that are 137 00:10:09,460 --> 00:10:11,170 normal to the edges of the cube. 138 00:10:13,910 --> 00:10:17,830 So all sorts of rotational symmetry. 139 00:10:17,830 --> 00:10:24,020 That, if you want a pattern, has a triangle that's on about 140 00:10:24,020 --> 00:10:28,600 each of the threefold axes. 141 00:10:28,600 --> 00:10:33,970 But the one that is up, points in the opposite orientation of 142 00:10:33,970 --> 00:10:37,760 the one that's on the threefold axis coming out of 143 00:10:37,760 --> 00:10:38,860 the other end of the cube. 144 00:10:38,860 --> 00:10:42,280 So you have one triangle that's like this, up. 145 00:10:42,280 --> 00:10:45,810 And another triangle like this that's on the other end of the 146 00:10:45,810 --> 00:10:48,570 threefold axis that points down into the blackboard. 147 00:10:53,630 --> 00:10:57,440 What can you add as other extenders? 148 00:10:57,440 --> 00:11:00,890 You can put in a mirror plane that is perpendicular to the 149 00:11:00,890 --> 00:11:02,480 fourfold axis. 150 00:11:02,480 --> 00:11:04,973 And that leaves everything unchanged. 151 00:11:09,740 --> 00:11:12,720 Now we've got a mirror plane passing through a fourfold 152 00:11:12,720 --> 00:11:15,370 axis, so we have to have mirror planes 153 00:11:15,370 --> 00:11:18,420 at 45 degree intervals. 154 00:11:18,420 --> 00:11:21,650 And that will create mirror planes there. 155 00:11:21,650 --> 00:11:24,050 This horizontal mirror plane goes through this fourfold 156 00:11:24,050 --> 00:11:28,370 axis, so there must be mirror planes at 45 degree intervals. 157 00:11:28,370 --> 00:11:30,040 So there's another one like this and 90 158 00:11:30,040 --> 00:11:31,930 degrees away as well. 159 00:11:31,930 --> 00:11:34,680 The fourfold axis is perpendicular to a mirror 160 00:11:34,680 --> 00:11:37,650 plane, as are these other twofold axes, so there's an 161 00:11:37,650 --> 00:11:39,140 inversion center. 162 00:11:39,140 --> 00:11:42,600 Here's a fourfold axis with a vertical mirror plane going 163 00:11:42,600 --> 00:11:42,930 through it. 164 00:11:42,930 --> 00:11:45,885 And it has to have a vertical mirror plane going like this 165 00:11:45,885 --> 00:11:47,090 at 45 degrees away. 166 00:11:47,090 --> 00:11:48,400 So that's the symmetry. 167 00:11:48,400 --> 00:11:52,910 This is the regular symmetry of a cube or an octahedron. 168 00:11:52,910 --> 00:11:55,880 And I will not, even if pressured, try to draw a 169 00:11:55,880 --> 00:11:59,870 pattern that conforms to that. 170 00:11:59,870 --> 00:12:01,990 We've got the fourfold axis 171 00:12:01,990 --> 00:12:04,990 perpendicular to a mirror plane. 172 00:12:04,990 --> 00:12:07,820 And this is called O sub h, O with a 173 00:12:07,820 --> 00:12:11,030 horizontal mirror plane. 174 00:12:11,030 --> 00:12:17,590 The fourfold axis in 4, 3, 2 has got a mirror plane 175 00:12:17,590 --> 00:12:18,670 perpendicular to it. 176 00:12:18,670 --> 00:12:20,860 The twofold axes all pick up mirror planes 177 00:12:20,860 --> 00:12:22,230 perpendicular to them. 178 00:12:22,230 --> 00:12:25,270 There's an inversion center at the point of intersection. 179 00:12:25,270 --> 00:12:28,520 So we label this axis a 3 bar axis. 180 00:12:28,520 --> 00:12:32,840 So that's O sub h, 4 over m, 3 bar, 2 over m. 181 00:12:32,840 --> 00:12:37,655 And this, as I say, is the symmetry of a regular cube or 182 00:12:37,655 --> 00:12:40,280 an octahedron. 183 00:12:40,280 --> 00:12:41,580 Nothing else we can do here. 184 00:12:41,580 --> 00:12:44,730 There's so many symmetry axes all over the place that 185 00:12:44,730 --> 00:12:48,730 there's no way we can snake in any diagonal mirror plane. 186 00:12:48,730 --> 00:12:53,070 Because there is no second axis of the same kind adjacent 187 00:12:53,070 --> 00:12:55,360 to either the twofold, the threefold, or 188 00:12:55,360 --> 00:12:56,400 the fourfold axis. 189 00:12:56,400 --> 00:12:59,230 So we can't put a mirror plane in here between the threefold 190 00:12:59,230 --> 00:13:00,680 and the twofold. 191 00:13:00,680 --> 00:13:03,245 We can't put a mirror plane in here between the fourfold and 192 00:13:03,245 --> 00:13:04,130 the twofold. 193 00:13:04,130 --> 00:13:05,060 So we're done. 194 00:13:05,060 --> 00:13:09,250 And this is the final and 30 second combination. 195 00:13:09,250 --> 00:13:14,082 So there are 32 crystallographic point groups. 196 00:13:14,082 --> 00:13:14,570 AUDIENCE: Professor? 197 00:13:14,570 --> 00:13:15,280 PROFESSOR: Yes, sir. 198 00:13:15,280 --> 00:13:22,084 AUDIENCE: So if m3m is a regular operation, are there 199 00:13:22,084 --> 00:13:23,334 symbols [INAUDIBLE]? 200 00:13:26,960 --> 00:13:29,190 PROFESSOR: O is the symbol for 4, 3, 2. 201 00:13:29,190 --> 00:13:32,690 And what we added as an extender is a mirror plane 202 00:13:32,690 --> 00:13:34,550 perpendicular to the fourfold axis. 203 00:13:34,550 --> 00:13:38,000 That is just one way of adding an extender. 204 00:13:38,000 --> 00:13:39,800 This is also Oi. 205 00:13:39,800 --> 00:13:42,690 If we add an inversion center, we get all this. 206 00:13:42,690 --> 00:13:46,090 And if you look at the ones have been honored by being 207 00:13:46,090 --> 00:13:51,250 designated by a symbol, the horizontal mirror plane takes 208 00:13:51,250 --> 00:13:53,230 the precedence. 209 00:13:53,230 --> 00:13:57,510 So for example, for 2 over m, 2 over m, 2 over m, you've got 210 00:13:57,510 --> 00:13:59,930 two different vertical mirror planes, and you've got one 211 00:13:59,930 --> 00:14:01,360 horizontal mirror plane. 212 00:14:01,360 --> 00:14:04,820 But it's called 2 over m, 2 over m, 2 over m. 213 00:14:04,820 --> 00:14:07,670 You can call it 2, 2, 2, m, m, m. 214 00:14:07,670 --> 00:14:12,050 That's also a possible symbol, but much more of a mouthful. 215 00:14:12,050 --> 00:14:15,720 There's some arbitrariness to the symbol because the 216 00:14:15,720 --> 00:14:18,900 arrangement of symmetry elements is what's real. 217 00:14:18,900 --> 00:14:22,180 And we decide how we want to devise a 218 00:14:22,180 --> 00:14:24,730 notation to label them. 219 00:14:24,730 --> 00:14:27,660 And as we've seen, there are two different people who 220 00:14:27,660 --> 00:14:30,690 adopted a different code for giving them names. 221 00:14:30,690 --> 00:14:32,870 So if it's rational and 222 00:14:32,870 --> 00:14:38,610 informative, it's a good notation. 223 00:14:38,610 --> 00:14:41,650 Interestingly, the international notation, on the 224 00:14:41,650 --> 00:14:44,520 one hand, and the Schoenflies notation, on the other, 225 00:14:44,520 --> 00:14:46,090 complimentary. 226 00:14:46,090 --> 00:14:48,280 The international notation tells you 227 00:14:48,280 --> 00:14:51,260 unambiguously what you have. 228 00:14:51,260 --> 00:14:56,310 So 2, 2, 2 over m, m, m tells you three orthogonal twofold 229 00:14:56,310 --> 00:14:58,400 axes if you remember what came out of [INAUDIBLE] 230 00:14:58,400 --> 00:14:59,320 construction. 231 00:14:59,320 --> 00:15:02,480 And perpendicular to each of them is a mirror plane. 232 00:15:02,480 --> 00:15:04,170 And that's what you've got. 233 00:15:04,170 --> 00:15:07,590 Schoenflies tells you how you derived it. 234 00:15:07,590 --> 00:15:13,120 You took D2, and that's the dihedral group 2, 2, 2, and 235 00:15:13,120 --> 00:15:16,340 you added an h, a horizontal mirror plane. 236 00:15:16,340 --> 00:15:17,490 And all hell broke loose. 237 00:15:17,490 --> 00:15:19,130 And you got mirror planes perpendicular to all the 238 00:15:19,130 --> 00:15:20,320 twofold axes. 239 00:15:20,320 --> 00:15:22,950 So there's a certain complementarity to the two 240 00:15:22,950 --> 00:15:24,950 different notations. 241 00:15:24,950 --> 00:15:29,870 And the people who do diffraction and 242 00:15:29,870 --> 00:15:34,760 crystallography for the most part follow the international 243 00:15:34,760 --> 00:15:37,520 notation, the Hermann-Mauguin notation. 244 00:15:37,520 --> 00:15:40,810 The people who do condensed matter physics use the 245 00:15:40,810 --> 00:15:42,340 Schoenflies notation. 246 00:15:42,340 --> 00:15:45,490 Because it's more inscrutable. 247 00:15:45,490 --> 00:15:47,670 And condensed matter physicists like to be 248 00:15:47,670 --> 00:15:52,190 inscrutable because it's how you gain respect. 249 00:15:52,190 --> 00:15:58,080 So both notation survive, but are more prevalent in some 250 00:15:58,080 --> 00:16:00,930 disciplines than in others and vice versa. 251 00:16:08,080 --> 00:16:11,590 Let's take a brief look ahead. 252 00:16:11,590 --> 00:16:13,610 What remains to be done? 253 00:16:13,610 --> 00:16:18,465 We now have the 32 crystallographic point groups. 254 00:16:21,280 --> 00:16:25,410 These are the way things can be arranged by symmetry about 255 00:16:25,410 --> 00:16:28,420 a fixed point in space. 256 00:16:28,420 --> 00:16:31,230 If we were to proceed in the same way that we did for the 257 00:16:31,230 --> 00:16:34,620 two-dimensional space groups, what we should do next is 258 00:16:34,620 --> 00:16:38,960 decide what sort of three-dimensional space 259 00:16:38,960 --> 00:16:42,760 lattices these symmetries will require. 260 00:16:42,760 --> 00:16:49,110 And then proceed to drop each of the point groups into each 261 00:16:49,110 --> 00:16:52,660 of the lattices that can accommodate them. 262 00:16:52,660 --> 00:16:56,060 And then use the tricks that we've used in two dimensions, 263 00:16:56,060 --> 00:16:59,920 take mirror planes and replace them by glide planes. 264 00:16:59,920 --> 00:17:03,130 And then having done that, we would take the mirror planes 265 00:17:03,130 --> 00:17:08,160 and the rotation axes and interweave them. 266 00:17:08,160 --> 00:17:12,060 And lest that job seem too daunting, let me point out 267 00:17:12,060 --> 00:17:15,760 that we already have 17 of the space groups. 268 00:17:15,760 --> 00:17:19,089 All we have to do is take the plane groups, take a 269 00:17:19,089 --> 00:17:21,579 translation that's perpendicular to the plane of 270 00:17:21,579 --> 00:17:24,914 the plane group, and let all the rotation axes and mirror 271 00:17:24,914 --> 00:17:28,054 planes extend up indefinitely in three dimensions parallel 272 00:17:28,054 --> 00:17:30,600 to that translation. 273 00:17:30,600 --> 00:17:33,740 So we already have 17 of the 274 00:17:33,740 --> 00:17:35,360 three-dimensional space groups. 275 00:17:35,360 --> 00:17:39,430 They look just like the plane groups except they extend in a 276 00:17:39,430 --> 00:17:43,020 direction that's perpendicular to the plane of our original 277 00:17:43,020 --> 00:17:45,030 two-dimensional group. 278 00:17:45,030 --> 00:17:48,260 So we're already a long way towards deriving a 279 00:17:48,260 --> 00:17:52,020 three-dimensional space group without really knowing it. 280 00:17:52,020 --> 00:17:55,130 But what I would like to consider next is the lattices 281 00:17:55,130 --> 00:17:57,870 that are required for three dimensions. 282 00:17:57,870 --> 00:18:02,160 And I've already given you how one can approach this problem. 283 00:18:02,160 --> 00:18:07,360 Take the lattices that have been required in the 284 00:18:07,360 --> 00:18:09,190 two-dimensional plane groups. 285 00:18:09,190 --> 00:18:13,280 And if the presence of a threefold axis and the base of 286 00:18:13,280 --> 00:18:19,510 the cell require, say, net that is hexagonal with a1 287 00:18:19,510 --> 00:18:29,160 equal to a2 identically in magnitude and exactly at 120 288 00:18:29,160 --> 00:18:30,945 degrees with respect to one another. 289 00:18:35,300 --> 00:18:39,770 Let's let the third translation be perpendicular 290 00:18:39,770 --> 00:18:42,170 to the base. 291 00:18:42,170 --> 00:18:44,430 If it's hexagonal, we have to have at least a 292 00:18:44,430 --> 00:18:47,690 threefold axis here. 293 00:18:47,690 --> 00:18:51,270 And we've found that adding a threefold axis to that net 294 00:18:51,270 --> 00:18:53,830 gave rise to two other threefold axes in the center 295 00:18:53,830 --> 00:18:55,200 of the triangle. 296 00:18:55,200 --> 00:18:59,740 So one lattice would be one in which the third translation, 297 00:18:59,740 --> 00:19:03,850 which we'll label c, is straight up. 298 00:19:08,530 --> 00:19:11,670 In other words, perpendicular to the net that we derived in 299 00:19:11,670 --> 00:19:14,010 two dimensions. 300 00:19:14,010 --> 00:19:17,320 What we will have as a constraint is that if we pick 301 00:19:17,320 --> 00:19:21,180 a certain translation that is not normal to the plane group, 302 00:19:21,180 --> 00:19:25,080 and let's say it terminates here in projection. 303 00:19:25,080 --> 00:19:28,130 So the third translation goes up and over. 304 00:19:28,130 --> 00:19:31,380 If there's a threefold axis at this end of the translation, 305 00:19:31,380 --> 00:19:34,010 there must be a threefold axis at the end of that 306 00:19:34,010 --> 00:19:35,280 translation. 307 00:19:35,280 --> 00:19:39,200 And that mucks everything up, and we no longer have a group. 308 00:19:39,200 --> 00:19:41,880 We can't have another threefold axis poking down 309 00:19:41,880 --> 00:19:45,520 through the plane of this two-dimensional plane group. 310 00:19:45,520 --> 00:19:50,300 But it is perfectly OK if the third translation moves this 311 00:19:50,300 --> 00:19:53,830 threefold axis and puts down a lattice point and another 312 00:19:53,830 --> 00:19:57,590 threefold axis directly over this one. 313 00:19:57,590 --> 00:20:00,270 Or alternatively, another choice for T3 314 00:20:00,270 --> 00:20:01,580 would be this one. 315 00:20:01,580 --> 00:20:04,980 That would put the lattice point in a threefold axis 316 00:20:04,980 --> 00:20:06,120 directly over this one. 317 00:20:06,120 --> 00:20:09,790 So again, we have threefold axes extending normal to the 318 00:20:09,790 --> 00:20:11,123 plane of my drawing. 319 00:20:11,123 --> 00:20:14,370 And I've created no threefold axes. 320 00:20:14,370 --> 00:20:19,220 So with a threefold axis, there are three potentially 321 00:20:19,220 --> 00:20:23,770 different space lattices, one in which the third translation 322 00:20:23,770 --> 00:20:26,760 is normal to the plane of the plane group, another one where 323 00:20:26,760 --> 00:20:28,900 the translation terminates over the 324 00:20:28,900 --> 00:20:31,200 other two twofold axes. 325 00:20:31,200 --> 00:20:33,290 And so this is the way in which one would proceed to 326 00:20:33,290 --> 00:20:36,950 derive the space lattices. 327 00:20:36,950 --> 00:20:38,710 Buerger does not do it this way. 328 00:20:38,710 --> 00:20:40,180 He doesn't look at the plane groups. 329 00:20:40,180 --> 00:20:45,150 He looks just at where the axes sit in the nets. 330 00:20:45,150 --> 00:20:47,150 No mirror planes whatsoever. 331 00:20:47,150 --> 00:20:48,610 And I'll return to the point. 332 00:20:48,610 --> 00:20:52,490 And the place where I think he's wrong is that if you add 333 00:20:52,490 --> 00:21:00,410 a twofold axis to a centered net, you get twofold axes in 334 00:21:00,410 --> 00:21:04,131 all of these locations. 335 00:21:04,131 --> 00:21:06,620 These are the twofold axes in C2mm. 336 00:21:10,000 --> 00:21:13,970 And if that's all you look at, there's no reason why this 337 00:21:13,970 --> 00:21:17,100 should be the plane group. 338 00:21:17,100 --> 00:21:20,800 And it looks as though you can make T3 terminate over this 339 00:21:20,800 --> 00:21:22,220 twofold axis. 340 00:21:22,220 --> 00:21:25,710 And that would give you a peculiar triple cell with 341 00:21:25,710 --> 00:21:28,680 three lattice points along the long diagonal of 342 00:21:28,680 --> 00:21:31,720 a rectangular net. 343 00:21:31,720 --> 00:21:35,490 And you say, wow, 15 space lattices. 344 00:21:35,490 --> 00:21:38,970 We're going to be famous, if not rich. 345 00:21:38,970 --> 00:21:43,620 You can't do that because this plane group can exist only if 346 00:21:43,620 --> 00:21:46,330 there are mirror planes in here like this. 347 00:21:46,330 --> 00:21:50,590 And then through these twofold axes, these have to be glides. 348 00:21:50,590 --> 00:21:54,820 So you can't take 2mm and put it on top of 2gg. 349 00:21:54,820 --> 00:21:56,830 It's impossible. 350 00:21:56,830 --> 00:22:00,050 So there are 14 space lattices. 351 00:22:00,050 --> 00:22:01,500 The 15th doesn't exist. 352 00:22:01,500 --> 00:22:04,000 But it looks as though it's possible in Buerger's 353 00:22:04,000 --> 00:22:06,040 treatment when he looks just at the placement of the 354 00:22:06,040 --> 00:22:09,180 rotation axes alone and not the location of any mirror 355 00:22:09,180 --> 00:22:13,040 planes that are in the plane groups. 356 00:22:13,040 --> 00:22:16,010 So that is where we're going to go next. 357 00:22:16,010 --> 00:22:19,000 And that is a process that actually is 358 00:22:19,000 --> 00:22:21,670 surprisingly simple. 359 00:22:21,670 --> 00:22:25,120 And we will get the space lattices very quickly. 360 00:22:25,120 --> 00:22:27,430 We'll get a number of space groups along the way, 361 00:22:27,430 --> 00:22:28,740 as we've just seen. 362 00:22:28,740 --> 00:22:32,100 And then we will cut to the bottom line and just look at 363 00:22:32,100 --> 00:22:36,700 how this information is tabulated in tables for you in 364 00:22:36,700 --> 00:22:39,230 a fashion that's analogous to the representation of the 365 00:22:39,230 --> 00:22:40,420 plane groups. 366 00:22:40,420 --> 00:22:41,260 So we're pretty close. 367 00:22:41,260 --> 00:22:44,325 We'll be wrapping things up in another two or three meetings. 368 00:22:46,830 --> 00:22:50,590 What I would like to do in the time that remains though, the 369 00:22:50,590 --> 00:22:54,496 point groups are very difficult to visualize unless 370 00:22:54,496 --> 00:22:59,130 you look at real crystals. 371 00:22:59,130 --> 00:23:03,640 So what I'm going to do is pass around a collection of 372 00:23:03,640 --> 00:23:05,360 models of actual crystals. 373 00:23:05,360 --> 00:23:09,870 I'm not sure I can tell you what every one of these models 374 00:23:09,870 --> 00:23:11,120 represents. 375 00:23:11,120 --> 00:23:12,740 These are curious things. 376 00:23:12,740 --> 00:23:15,980 I would beg you not to drop them on one 377 00:23:15,980 --> 00:23:19,030 of the sharp corners. 378 00:23:19,030 --> 00:23:20,210 These are made out of wood. 379 00:23:20,210 --> 00:23:25,200 They're made out of pear wood in the Black Forest by elves. 380 00:23:25,200 --> 00:23:28,230 No, that last part is not true. 381 00:23:28,230 --> 00:23:30,700 But they are incredibly expensive. 382 00:23:30,700 --> 00:23:32,880 Because the angles in these things have 383 00:23:32,880 --> 00:23:34,780 to be exactly right. 384 00:23:34,780 --> 00:23:38,140 So they are made on the same sort of machine that you use 385 00:23:38,140 --> 00:23:40,480 for faceting diamonds. 386 00:23:40,480 --> 00:23:43,380 It's something that has two degrees of freedom, so you can 387 00:23:43,380 --> 00:23:47,220 get a surface that you want on the material exactly parallel 388 00:23:47,220 --> 00:23:48,690 to a grinding surface. 389 00:23:48,690 --> 00:23:52,150 And then you have a provision for advancing it normal to 390 00:23:52,150 --> 00:23:54,010 that direction by a controlled amount. 391 00:23:54,010 --> 00:23:57,130 If it were not precise, you would not see sharp edges and 392 00:23:57,130 --> 00:23:58,200 sharp corners. 393 00:23:58,200 --> 00:24:02,000 So these are incredibly expensive. 394 00:24:02,000 --> 00:24:05,310 I, therefore, ask it that you hold them 395 00:24:05,310 --> 00:24:06,625 tightly with both hands. 396 00:24:06,625 --> 00:24:08,170 Don't drop them on the floor. 397 00:24:08,170 --> 00:24:10,420 And I'll take around a couple of handfuls of these. 398 00:24:10,420 --> 00:24:12,660 I don't know if I have enough for everyone. 399 00:24:12,660 --> 00:24:15,570 With the tables of the point groups in front of you, why 400 00:24:15,570 --> 00:24:19,040 don't you try to identify the point group that's possible in 401 00:24:19,040 --> 00:24:20,290 each of these models. 402 00:24:26,820 --> 00:24:27,640 This is a big one. 403 00:24:27,640 --> 00:24:29,487 So I'm sure you won't drop that one. 404 00:24:45,400 --> 00:24:47,770 I don't think I have enough for everyone, so I'll start 405 00:24:47,770 --> 00:24:49,530 passing them out to every other person. 406 00:24:49,530 --> 00:24:51,820 Maybe you can look on with the person adjacent to you. 407 00:25:00,269 --> 00:25:01,519 Take one of these. 408 00:25:18,658 --> 00:25:19,908 Pass those over. 409 00:25:28,598 --> 00:25:29,848 Here you go. 410 00:25:43,600 --> 00:25:45,970 I gave you that just to be mean. 411 00:25:45,970 --> 00:25:48,167 That is an example of something that's called a 412 00:25:48,167 --> 00:25:50,010 twinned crystal. 413 00:25:50,010 --> 00:25:51,770 You can't have reentrant faces. 414 00:25:51,770 --> 00:25:54,140 Yeah, that's actually two crystals that are intergrown. 415 00:25:58,880 --> 00:26:00,840 AUDIENCE: So we can't identify the [INAUDIBLE]? 416 00:26:00,840 --> 00:26:02,090 PROFESSOR: You can look. 417 00:26:02,090 --> 00:26:05,470 Block one of them out in your mind, and try to imagine what 418 00:26:05,470 --> 00:26:06,310 it would look like. 419 00:26:06,310 --> 00:26:06,580 AUDIENCE: [INAUDIBLE]. 420 00:26:06,580 --> 00:26:07,830 PROFESSOR: Yeah. 421 00:26:16,090 --> 00:26:17,990 Here's an example of another one. 422 00:26:17,990 --> 00:26:20,220 That's actually gypsum. 423 00:26:20,220 --> 00:26:23,110 And that's a very common twin in gypsum. 424 00:26:23,110 --> 00:26:26,580 And if you could take this and rotate it-- 425 00:26:26,580 --> 00:26:27,810 this one doesn't rotate-- 426 00:26:27,810 --> 00:26:31,660 rotate it 90 degrees, then you would have a crystal that had 427 00:26:31,660 --> 00:26:34,382 a parallelogram shape. 428 00:26:34,382 --> 00:26:37,490 And actually that's been rotated by 90 429 00:26:37,490 --> 00:26:39,790 degrees about an axis. 430 00:26:39,790 --> 00:26:41,040 That's within the base. 431 00:26:45,635 --> 00:26:50,366 AUDIENCE: No, no, I think that's just 4m or 4 over m. 432 00:26:50,366 --> 00:26:52,060 AUDIENCE: Yeah, there's no mirror there. 433 00:26:52,060 --> 00:26:54,720 That's a 4 bar, I think. 434 00:26:54,720 --> 00:26:56,570 AUDIENCE: Oh, is that the inversion? 435 00:26:56,570 --> 00:26:59,940 PROFESSOR: There's no-- neither. 436 00:26:59,940 --> 00:27:01,330 I am a touchy-feely guy. 437 00:27:01,330 --> 00:27:04,000 If I see something up here, I look down here and try to feel 438 00:27:04,000 --> 00:27:06,020 something that's parallel to it. 439 00:27:06,020 --> 00:27:07,350 But you're right, you're right. 440 00:27:07,350 --> 00:27:09,520 If I rotate 90 degrees-- 441 00:27:09,520 --> 00:27:12,650 remember rotoinversion is the same as rotoreflection. 442 00:27:12,650 --> 00:27:15,730 So I could rotate to here and then reflect down, 443 00:27:15,730 --> 00:27:16,980 and I get this one. 444 00:27:19,320 --> 00:27:20,210 AUDIENCE: So it's 4 bar. 445 00:27:20,210 --> 00:27:22,080 PROFESSOR: It's 4 bar, yeah. 446 00:27:22,080 --> 00:27:24,470 And you see in the cross section, it is square. 447 00:27:24,470 --> 00:27:28,260 There's a little square on top here, so there is a fourfold 448 00:27:28,260 --> 00:27:30,440 aspect to it when you look at planes in a special 449 00:27:30,440 --> 00:27:30,915 orientation. 450 00:27:30,915 --> 00:27:33,290 AUDIENCE: I have a question about 4 bar. 451 00:27:33,290 --> 00:27:38,050 4 bar in planes are twofold rotation. 452 00:27:38,050 --> 00:27:39,870 PROFESSOR: A twofold pure rotation. 453 00:27:39,870 --> 00:27:43,520 Probably the best example of 4 bar is a tetrahedron. 454 00:27:43,520 --> 00:27:47,350 Two faces on top, two faces twisted 90 degrees, and 455 00:27:47,350 --> 00:27:48,110 inverted down. 456 00:27:48,110 --> 00:27:50,006 AUDIENCE: So that means an 8 bar would be a 457 00:27:50,006 --> 00:27:50,954 fourfold proper rotation. 458 00:27:50,954 --> 00:27:53,798 So why cannot I put an 8 bar in the point group? 459 00:27:53,798 --> 00:27:57,510 Because it's only a fourfold rotation, and that's allowed. 460 00:27:57,510 --> 00:28:00,720 PROFESSOR: The reason is that there's no origin to 461 00:28:00,720 --> 00:28:01,650 translations. 462 00:28:01,650 --> 00:28:06,160 So if you had translations in an 8 bar crystal, you would 463 00:28:06,160 --> 00:28:11,940 have four translations that were repeated by rotation and 464 00:28:11,940 --> 00:28:16,920 then another translation 45 degrees and inverted down. 465 00:28:16,920 --> 00:28:19,760 But you can assemble those at this same point. 466 00:28:19,760 --> 00:28:23,710 So you would have eight translations 45 degrees apart. 467 00:28:23,710 --> 00:28:25,240 And that's impossible in a lattice. 468 00:28:30,657 --> 00:28:31,480 Need another one? 469 00:28:31,480 --> 00:28:32,980 AUDIENCE: I'm sorry. 470 00:28:32,980 --> 00:28:34,475 PROFESSOR: Do you need another one to look at? 471 00:28:34,475 --> 00:28:35,327 AUDIENCE: No, I'm good. 472 00:28:35,327 --> 00:28:36,577 PROFESSOR: OK. 473 00:28:39,620 --> 00:28:41,770 I can't have you just sitting there doing nothing. 474 00:28:46,153 --> 00:28:50,049 AUDIENCE: So is this half of this [INAUDIBLE] material? 475 00:28:50,049 --> 00:28:52,760 PROFESSOR: Yeah, that's twinned by a 180 degree 476 00:28:52,760 --> 00:28:54,872 rotation on the 1, 1, 1 plane. 477 00:28:54,872 --> 00:28:57,332 AUDIENCE: Oh, yeah, the 1, 1, 1 [INAUDIBLE]. 478 00:29:00,776 --> 00:29:02,050 PROFESSOR: Yeah. 479 00:29:02,050 --> 00:29:03,960 That is actually half of an octahedron. 480 00:29:12,590 --> 00:29:14,660 This is a-- whoops, no, it's not. 481 00:29:14,660 --> 00:29:15,910 Yes, it is. 482 00:29:25,716 --> 00:29:28,520 No, I think that's just the threefold-- 483 00:29:28,520 --> 00:29:30,090 3, 3, 3. 484 00:29:33,560 --> 00:29:35,950 3 bar, 2 over m. 485 00:29:35,950 --> 00:29:39,730 And it's been rotated by 180 degrees, which is not a 486 00:29:39,730 --> 00:29:41,450 symmetry transformation. 487 00:29:41,450 --> 00:29:45,535 So the 3 bar is rotated 180 degrees. 488 00:29:45,535 --> 00:29:47,500 AUDIENCE: That's a 3 bar, 2 over m? 489 00:29:47,500 --> 00:29:47,920 PROFESSOR: I think so. 490 00:29:47,920 --> 00:29:52,460 I think it's a 3 bar here, a face here, and a face down 491 00:29:52,460 --> 00:29:56,100 below, inverted, 60 degrees away. 492 00:29:56,100 --> 00:29:58,970 And they're mirror planes. 493 00:29:58,970 --> 00:30:04,785 And there two full axes perpendicular to those mirror 494 00:30:04,785 --> 00:30:08,110 planes, I do believe. 495 00:30:08,110 --> 00:30:10,490 AUDIENCE: I don't see the twofold axes. 496 00:30:10,490 --> 00:30:12,830 PROFESSOR: You're only seeing half the crystal, and I think 497 00:30:12,830 --> 00:30:14,080 that's the reason why. 498 00:30:21,170 --> 00:30:23,050 This is the mirror plane here. 499 00:30:23,050 --> 00:30:26,585 And this edge is parallel to that mirror plane. 500 00:30:26,585 --> 00:30:29,060 And there's a twofold axis that comes out of the middle 501 00:30:29,060 --> 00:30:31,230 of that edge. 502 00:30:31,230 --> 00:30:32,350 You can't see it. 503 00:30:32,350 --> 00:30:33,940 That edge. 504 00:30:33,940 --> 00:30:37,360 And the mirror plane is exactly parallel to that. 505 00:30:37,360 --> 00:30:39,868 It's tough to see because you're only seeing half of it. 506 00:30:39,868 --> 00:30:40,836 AUDIENCE: [INAUDIBLE]. 507 00:30:40,836 --> 00:30:41,320 PROFESSOR: Yeah. 508 00:30:41,320 --> 00:30:43,760 AUDIENCE: This [INAUDIBLE]? 509 00:30:43,760 --> 00:30:44,300 PROFESSOR: Yeah. 510 00:30:44,300 --> 00:30:46,840 This is a fourfold axis 511 00:30:46,840 --> 00:30:49,015 perpendicular to a mirror plane. 512 00:30:49,015 --> 00:30:52,420 There's a threefold axis coming out here and a twofold 513 00:30:52,420 --> 00:30:54,110 axis coming out here. 514 00:30:54,110 --> 00:30:56,740 And there is a mirror plane perpendicular to the fourfold 515 00:30:56,740 --> 00:31:01,180 axis and perpendicular to the twofold axis also under the 516 00:31:01,180 --> 00:31:01,490 mirror plane. 517 00:31:01,490 --> 00:31:05,876 So this is 4 over m, 3 bar, 2 over m. 518 00:31:05,876 --> 00:31:07,180 AUDIENCE: [INAUDIBLE]. 519 00:31:07,180 --> 00:31:08,780 PROFESSOR: That's the rotational symmetry. 520 00:31:08,780 --> 00:31:10,335 But you get all sorts of mirror planes 521 00:31:10,335 --> 00:31:12,120 coming through here. 522 00:31:12,120 --> 00:31:15,870 So really, it's this one. 523 00:31:15,870 --> 00:31:18,490 Mirror plane this way, mirror plane this way, 524 00:31:18,490 --> 00:31:19,996 mirror plane this way. 525 00:31:23,650 --> 00:31:30,200 So if we set it up relative to this, there's the fourfold 526 00:31:30,200 --> 00:31:34,700 coming out here, here's the fourfold coming out here. 527 00:31:34,700 --> 00:31:40,650 And there are mirror planes this way and also this way, 45 528 00:31:40,650 --> 00:31:41,900 degrees away. 529 00:31:50,915 --> 00:31:52,165 Nothing to do? 530 00:31:57,125 --> 00:31:58,930 Can I steal one that you're not working with? 531 00:31:58,930 --> 00:31:59,480 There's a-- 532 00:31:59,480 --> 00:32:00,730 AUDIENCE: This one is [INAUDIBLE]. 533 00:32:13,420 --> 00:32:18,710 PROFESSOR: That's 2, 3 with no mirror planes. 534 00:32:18,710 --> 00:32:22,790 Oh, no, no, that's not true. 535 00:32:22,790 --> 00:32:29,224 No, that is 4 bar, 2m, believe it or not. 536 00:32:36,610 --> 00:32:40,483 Look at the three different directions. 537 00:32:43,700 --> 00:32:45,620 These two corners are the same. 538 00:32:45,620 --> 00:32:49,790 There's a little, tiny line segment there. 539 00:32:49,790 --> 00:32:52,192 AUDIENCE: Yeah, so there is a mirror plane right here? 540 00:32:52,192 --> 00:32:54,240 PROFESSOR: Yeah, but there's no mirror plane going through 541 00:32:54,240 --> 00:32:56,010 the other edges. 542 00:32:56,010 --> 00:33:00,990 So there's a mirror plane here, mirror plane here. 543 00:33:00,990 --> 00:33:05,390 And then there is twofold axes coming out of this little 544 00:33:05,390 --> 00:33:06,970 straight line segment here. 545 00:33:06,970 --> 00:33:07,466 AUDIENCE: Really? 546 00:33:07,466 --> 00:33:07,962 PROFESSOR: Yeah. 547 00:33:07,962 --> 00:33:10,120 And that's different from this. 548 00:33:10,120 --> 00:33:13,544 So these two edges are the same. 549 00:33:13,544 --> 00:33:17,265 AUDIENCE: Where is the principle axis? 550 00:33:17,265 --> 00:33:19,550 PROFESSOR: The principle axis would be this one. 551 00:33:19,550 --> 00:33:20,240 AUDIENCE: This one? 552 00:33:20,240 --> 00:33:21,606 PROFESSOR: Yeah. 553 00:33:21,606 --> 00:33:27,820 AUDIENCE: So basically, I have this mirror plane right here. 554 00:33:27,820 --> 00:33:35,220 Then twofold axis in the middle of the circle. 555 00:33:35,220 --> 00:33:44,625 So now here, where is it? 556 00:33:44,625 --> 00:33:46,305 PROFESSOR: Well, it's hexagonal. 557 00:33:46,305 --> 00:33:47,890 So you want to back up a little bit. 558 00:33:53,236 --> 00:33:55,180 Right there. 559 00:33:55,180 --> 00:33:57,124 AUDIENCE: This one? 560 00:33:57,124 --> 00:33:58,374 PROFESSOR: No, sorry. 561 00:34:02,956 --> 00:34:04,920 This one. 562 00:34:04,920 --> 00:34:07,590 Threefold axis, twofold. 563 00:34:07,590 --> 00:34:09,538 That's perpendicular to a mirror plane. 564 00:34:09,538 --> 00:34:11,410 AUDIENCE: So here's my mirror plane. 565 00:34:14,218 --> 00:34:16,730 PROFESSOR: Coming right out of this little edge here. 566 00:34:16,730 --> 00:34:18,554 AUDIENCE: Let's say like this. 567 00:34:18,554 --> 00:34:19,466 No mirror plane? 568 00:34:19,466 --> 00:34:21,750 PROFESSOR: No. 569 00:34:21,750 --> 00:34:23,630 AUDIENCE: Which one is this one? 570 00:34:23,630 --> 00:34:25,679 It's like this one. 571 00:34:25,679 --> 00:34:26,900 PROFESSOR: Let's put it right here. 572 00:34:26,900 --> 00:34:29,726 Here is-- 573 00:34:29,726 --> 00:34:30,699 AUDIENCE: Like this? 574 00:34:30,699 --> 00:34:34,219 PROFESSOR: No, you've got to get it up like this. 575 00:34:34,219 --> 00:34:35,290 OK. 576 00:34:35,290 --> 00:34:36,830 There is a mirror plane. 577 00:34:36,830 --> 00:34:41,400 AUDIENCE: No, this is not. 578 00:34:41,400 --> 00:34:43,211 No, the mirror plane is right here. 579 00:34:43,211 --> 00:34:45,920 PROFESSOR: Yeah, OK. 580 00:34:45,920 --> 00:34:48,584 It goes up like-- 581 00:34:48,584 --> 00:34:50,500 oh. 582 00:34:50,500 --> 00:34:53,969 Let me get it set up here. 583 00:34:53,969 --> 00:34:54,300 OK. 584 00:34:54,300 --> 00:34:55,830 There is the mirror plane. 585 00:34:55,830 --> 00:34:59,620 This face is different from the others. 586 00:34:59,620 --> 00:35:03,452 AUDIENCE: And the twofold axes are on the edges. 587 00:35:03,452 --> 00:35:04,890 PROFESSOR: Yeah. 588 00:35:04,890 --> 00:35:07,437 AUDIENCE: So basically, there are three 589 00:35:07,437 --> 00:35:09,345 for one mirror plane. 590 00:35:12,210 --> 00:35:14,420 PROFESSOR: Here are the mirror planes coming through this 591 00:35:14,420 --> 00:35:16,760 way, this way, this way. 592 00:35:16,760 --> 00:35:20,080 They're 60 degrees apart. 593 00:35:20,080 --> 00:35:22,520 AUDIENCE: You say there are three mirror planes? 594 00:35:22,520 --> 00:35:25,320 PROFESSOR: Yeah, this one, this one, and this one. 595 00:35:25,320 --> 00:35:26,470 AUDIENCE: I don't agree. 596 00:35:26,470 --> 00:35:28,696 This one is no mirror plane. 597 00:35:28,696 --> 00:35:30,055 PROFESSOR: You're right. 598 00:35:30,055 --> 00:35:31,751 You're right. 599 00:35:31,751 --> 00:35:35,030 AUDIENCE: There is only one when you look from above. 600 00:35:35,030 --> 00:35:38,445 But there are two for just [INAUDIBLE] 601 00:35:38,445 --> 00:35:40,870 45 degrees-- or, not 45 degrees. 602 00:35:40,870 --> 00:35:42,120 That's a triangle. 603 00:35:52,040 --> 00:35:53,410 PROFESSOR: It's a terrible one. 604 00:36:17,218 --> 00:36:19,450 AUDIENCE: Because if we decide that the top is one, the 605 00:36:19,450 --> 00:36:21,020 principle-- 606 00:36:21,020 --> 00:36:22,420 PROFESSOR: There's a twofold axis. 607 00:36:22,420 --> 00:36:26,220 That's a mirror plane, and that's a mirror plane. 608 00:36:26,220 --> 00:36:29,140 And it looks like it is 2nn. 609 00:36:34,540 --> 00:36:36,761 I think it's 2nn. 610 00:36:36,761 --> 00:36:39,647 AUDIENCE: OK. 611 00:36:39,647 --> 00:36:40,609 There's nothing further? 612 00:36:40,609 --> 00:36:41,859 PROFESSOR: Nothing further. 613 00:36:45,900 --> 00:36:47,090 Got that one? 614 00:36:47,090 --> 00:36:49,440 AUDIENCE: 4 bar 2m? 615 00:36:49,440 --> 00:36:51,320 I had to look to find a twofold axis. 616 00:36:51,320 --> 00:36:52,270 That was the tricky part. 617 00:36:52,270 --> 00:36:54,388 PROFESSOR: No, no threefold axis. 618 00:36:54,388 --> 00:36:55,296 AUDIENCE: Well, twofold. 619 00:36:55,296 --> 00:36:57,112 PROFESSOR: Yeah. 620 00:36:57,112 --> 00:36:59,904 AUDIENCE: Finding those was [INAUDIBLE]. 621 00:36:59,904 --> 00:37:00,840 PROFESSOR: That's it. 622 00:37:00,840 --> 00:37:02,480 AUDIENCE: I can see the mirror planes. 623 00:37:02,480 --> 00:37:03,790 PROFESSOR: That's the twofold axis. 624 00:37:03,790 --> 00:37:06,730 AUDIENCE: Yeah. 625 00:37:06,730 --> 00:37:09,710 PROFESSOR: And this two on top and two underneath skewed by 626 00:37:09,710 --> 00:37:11,730 90 degrees, that's a 4 bar. 627 00:37:11,730 --> 00:37:13,090 So that's 4 bar 2m. 628 00:37:19,382 --> 00:37:22,300 You have nothing to do? 629 00:37:22,300 --> 00:37:23,680 Oh, you're talking with them, OK. 630 00:37:34,540 --> 00:37:37,650 AUDIENCE: What is this one actually? 631 00:37:37,650 --> 00:37:41,030 There is 4, 4, 2, 4, 3, 4, 3. 632 00:37:41,030 --> 00:37:43,465 PROFESSOR: Right, so it's cubic. 633 00:37:43,465 --> 00:37:49,540 And mirror planes are going down through the twofold axis. 634 00:37:49,540 --> 00:37:52,190 So that's enough to tell you that it is this. 635 00:37:55,320 --> 00:37:57,810 So let's set it up here. 636 00:37:57,810 --> 00:38:00,860 Fourfold axes are coming out of the points. 637 00:38:00,860 --> 00:38:03,210 So you've got mirror planes this way, 638 00:38:03,210 --> 00:38:05,260 mirror point 45 degrees. 639 00:38:05,260 --> 00:38:07,240 Here is this twofold axis. 640 00:38:07,240 --> 00:38:09,710 Here is this twofold axis. 641 00:38:09,710 --> 00:38:12,360 And here are the twofold axes that are in the same plane. 642 00:38:12,360 --> 00:38:13,810 And here's the other fourfold axis. 643 00:38:13,810 --> 00:38:15,060 AUDIENCE: I see. 644 00:38:24,560 --> 00:38:26,590 That one's not fair. 645 00:38:26,590 --> 00:38:34,180 This is actually two crystals that are grown together by an 646 00:38:34,180 --> 00:38:36,780 operation that is not a symmetry 647 00:38:36,780 --> 00:38:38,320 element of the crystal. 648 00:38:38,320 --> 00:38:40,430 And actually these two crystals have 649 00:38:40,430 --> 00:38:44,300 symmetry 2 over m. 650 00:38:44,300 --> 00:38:47,460 And they are rotated relative to one another by 651 00:38:47,460 --> 00:38:49,780 a 180 degree rotation. 652 00:38:49,780 --> 00:38:51,360 It's not a symmetry element. 653 00:38:51,360 --> 00:38:54,440 So this is something that's called a twinned crystal. 654 00:38:54,440 --> 00:38:54,740 AUDIENCE: Twinned? 655 00:38:54,740 --> 00:38:56,960 PROFESSOR: Twinned crystal. 656 00:38:56,960 --> 00:38:58,170 So it's really two of them. 657 00:38:58,170 --> 00:39:02,580 And if I could twist this one around so that the crystal 658 00:39:02,580 --> 00:39:04,500 continued on in that direction, it would be 659 00:39:04,500 --> 00:39:07,890 something that had a lozenge-like shape. 660 00:39:07,890 --> 00:39:09,340 So I should put that one away. 661 00:39:09,340 --> 00:39:11,170 That's only confusing people. 662 00:39:11,170 --> 00:39:12,250 AUDIENCE: Professor Wuensch? 663 00:39:12,250 --> 00:39:12,620 PROFESSOR: Yes, sir. 664 00:39:12,620 --> 00:39:13,870 AUDIENCE: Is that a 6 over mmm? 665 00:39:17,350 --> 00:39:21,090 PROFESSOR: Yes, that's 6 over m, 2 over m, 2 over m. 666 00:39:21,090 --> 00:39:22,410 Six folds. 667 00:39:22,410 --> 00:39:24,730 Two kinds of twofold, one out of the edge, 668 00:39:24,730 --> 00:39:26,880 one out of the corner. 669 00:39:26,880 --> 00:39:27,810 And a mirror plane 670 00:39:27,810 --> 00:39:30,810 perpendicular to each of those. 671 00:39:30,810 --> 00:39:32,650 6 over m, 2 over m, 2 over m. 672 00:39:32,650 --> 00:39:36,386 AUDIENCE: So that's the same thing [INAUDIBLE]. 673 00:39:40,494 --> 00:39:42,979 So the 6 over m comes from this way. 674 00:39:42,979 --> 00:39:45,464 And then the 2 over m is there. 675 00:39:45,464 --> 00:39:47,452 AUDIENCE: Yeah, so it's [INAUDIBLE]. 676 00:39:47,452 --> 00:39:48,702 AUDIENCE: Yeah, exactly. 677 00:39:52,440 --> 00:39:53,150 PROFESSOR: Let me put that one away. 678 00:39:53,150 --> 00:39:54,700 It's just confusing people. 679 00:39:54,700 --> 00:39:55,460 AUDIENCE: Oh, that one was easy. 680 00:39:55,460 --> 00:39:56,800 PROFESSOR: That's a twinned crystal. 681 00:39:56,800 --> 00:39:57,240 It was easy? 682 00:39:57,240 --> 00:39:57,920 You found it easy? 683 00:39:57,920 --> 00:39:59,610 AUDIENCE: Yeah, so 3 over m, right? 684 00:40:03,270 --> 00:40:04,520 PROFESSOR: Yeah, I guess. 685 00:40:10,120 --> 00:40:12,850 This looks as though it might be the corner of an octahedron 686 00:40:12,850 --> 00:40:13,920 because you don't see much of it. 687 00:40:13,920 --> 00:40:17,490 But these two faces and these two faces are not related by a 688 00:40:17,490 --> 00:40:18,740 fourfold rotation. 689 00:40:20,600 --> 00:40:22,370 AUDIENCE: Well, either way, there wouldn't be anywhere to 690 00:40:22,370 --> 00:40:23,330 put a fourfold rotation. 691 00:40:23,330 --> 00:40:24,760 PROFESSOR: Yeah, right. 692 00:40:24,760 --> 00:40:27,590 So if you look at the whole thing, this is actually two 693 00:40:27,590 --> 00:40:29,690 intergrown crystals when you see this sort 694 00:40:29,690 --> 00:40:31,110 of reentrant angle. 695 00:40:31,110 --> 00:40:36,570 This is two crystals grown together by an operation which 696 00:40:36,570 --> 00:40:39,252 is not a symmetry operation. 697 00:40:39,252 --> 00:40:40,797 So this is something that's called a 698 00:40:40,797 --> 00:40:42,420 twin, a twinned crystal. 699 00:40:42,420 --> 00:40:43,830 AUDIENCE: But the symmetry is 700 00:40:43,830 --> 00:40:45,650 nevertheless 3 over m, correct? 701 00:40:45,650 --> 00:40:46,330 PROFESSOR: Of the whole thing? 702 00:40:46,330 --> 00:40:47,550 Yeah, yeah, of the whole thing. 703 00:40:47,550 --> 00:40:50,100 AUDIENCE: And this is just 3. 704 00:40:50,100 --> 00:40:51,750 PROFESSOR: No, this is 3mm. 705 00:40:56,080 --> 00:40:59,075 There's a mirror plane down there and a 706 00:40:59,075 --> 00:41:00,210 mirror plane down here. 707 00:41:00,210 --> 00:41:01,020 AUDIENCE: You're right. 708 00:41:01,020 --> 00:41:03,980 PROFESSOR: No twofold axis because this end is different 709 00:41:03,980 --> 00:41:06,100 from this end. 710 00:41:06,100 --> 00:41:07,580 AUDIENCE: So this is a regular solid, right? 711 00:41:07,580 --> 00:41:09,050 What is it called? 712 00:41:09,050 --> 00:41:12,188 PROFESSOR: This is a rhombic dodecahedron. 713 00:41:12,188 --> 00:41:14,164 AUDIENCE: Oh, the one you were talking about the other day. 714 00:41:14,164 --> 00:41:14,660 PROFESSOR: Yeah. 715 00:41:14,660 --> 00:41:19,640 This is 4, 3, 2. 716 00:41:19,640 --> 00:41:21,890 And then mirror planes too all over the place. 717 00:41:21,890 --> 00:41:23,630 AUDIENCE: So all of these solids do not have to be 718 00:41:23,630 --> 00:41:25,840 members of this list, right? 719 00:41:25,840 --> 00:41:28,460 Because these are finite objects. 720 00:41:28,460 --> 00:41:30,610 They can be [? twelvefold ?] and-- 721 00:41:30,610 --> 00:41:31,290 PROFESSOR: They could be. 722 00:41:31,290 --> 00:41:34,110 These actually, in fact, are models of 723 00:41:34,110 --> 00:41:35,820 real crystalline materials. 724 00:41:35,820 --> 00:41:39,490 And they're made with the faces and the relative sizes 725 00:41:39,490 --> 00:41:41,835 that these minerals actually have. 726 00:41:41,835 --> 00:41:43,760 AUDIENCE: OK. 727 00:41:43,760 --> 00:41:46,020 PROFESSOR: So somewhere floating around is something 728 00:41:46,020 --> 00:41:48,830 that's very characteristically quartz. 729 00:41:48,830 --> 00:41:52,095 And there's one that's a flat one with a reentrant angle in 730 00:41:52,095 --> 00:41:53,345 it, that's gypsum. 731 00:41:59,170 --> 00:42:01,080 AUDIENCE: [INAUDIBLE]. 732 00:42:01,080 --> 00:42:03,820 PROFESSOR: Yeah, that's fourfold. 733 00:42:03,820 --> 00:42:05,560 Twofold here. 734 00:42:05,560 --> 00:42:07,900 This goes into this. 735 00:42:07,900 --> 00:42:10,740 No mirror planes because these things are inclined. 736 00:42:10,740 --> 00:42:15,942 So it's 4, 2, 2. 737 00:42:15,942 --> 00:42:17,192 Good. 738 00:42:19,770 --> 00:42:21,430 It's easy when you know what to look for. 739 00:42:21,430 --> 00:42:24,010 When you say, they are only a few possibilities. 740 00:42:24,010 --> 00:42:27,630 And if I see a fourfold axis in it, that narrows it down to 741 00:42:27,630 --> 00:42:28,880 two or three. 742 00:42:43,385 --> 00:42:43,870 AUDIENCE: Like that. 743 00:42:43,870 --> 00:42:48,525 That's an inversion center running through here. 744 00:42:48,525 --> 00:42:51,614 And there's no rotation there [INAUDIBLE]. 745 00:42:51,614 --> 00:42:53,606 AUDIENCE: [INAUDIBLE] here? 746 00:42:53,606 --> 00:42:55,598 Here through this way? 747 00:42:58,586 --> 00:43:02,072 AUDIENCE: No, because this doesn't map [INAUDIBLE]. 748 00:43:05,060 --> 00:43:09,044 Does that map that by a mirror? 749 00:43:09,044 --> 00:43:11,036 The mirror [INAUDIBLE]? 750 00:43:11,036 --> 00:43:13,526 AUDIENCE: It cuts across this way. 751 00:43:13,526 --> 00:43:16,514 AUDIENCE: Yeah, I could buy that. 752 00:43:19,530 --> 00:43:23,070 PROFESSOR: You've reached a consensus? 753 00:43:23,070 --> 00:43:24,690 AUDIENCE: We've got a mirror, and we've got an inversion. 754 00:43:24,690 --> 00:43:27,240 But we're not sure what else we've got. 755 00:43:27,240 --> 00:43:30,471 AUDIENCE: While I like the inversion, I'd 756 00:43:30,471 --> 00:43:31,721 also like the mirror. 757 00:43:39,990 --> 00:43:41,240 I don't see the mirror. 758 00:43:49,492 --> 00:43:51,977 All I see is a really an inversion axis there. 759 00:43:51,977 --> 00:43:55,960 I think I'm missing that mirror that you're saying. 760 00:43:55,960 --> 00:43:58,610 PROFESSOR: There's a mirror that goes down this way. 761 00:43:58,610 --> 00:44:00,800 And there's a twofold axis here. 762 00:44:03,395 --> 00:44:04,790 AUDIENCE: OK, that was the one we missed. 763 00:44:04,790 --> 00:44:06,570 PROFESSOR: So that's 2 over m. 764 00:44:06,570 --> 00:44:09,340 AUDIENCE: Over m. 765 00:44:09,340 --> 00:44:10,860 And the inversion just falls out. 766 00:44:10,860 --> 00:44:13,050 PROFESSOR: Yeah, that's in there too. 767 00:44:13,050 --> 00:44:15,980 Inversions are nice. 768 00:44:15,980 --> 00:44:17,340 You can feel inversions. 769 00:44:17,340 --> 00:44:20,160 You put your finger on one face and your finger on the 770 00:44:20,160 --> 00:44:21,380 other face. 771 00:44:21,380 --> 00:44:22,630 [INTERPOSING VOICES] 772 00:44:39,092 --> 00:44:39,584 PROFESSOR: Get it? 773 00:44:39,584 --> 00:44:40,950 AUDIENCE: So this is this one, right? 774 00:44:40,950 --> 00:44:43,890 PROFESSOR: Yes, very good. 775 00:44:43,890 --> 00:44:47,865 And that is a silicate mineral called garnet. 776 00:44:47,865 --> 00:44:48,820 AUDIENCE: What's that? 777 00:44:48,820 --> 00:44:50,060 PROFESSOR: Garnet. 778 00:44:50,060 --> 00:44:54,030 It's actually a silicate mineral, which is used as a 779 00:44:54,030 --> 00:44:55,280 gemstone sometimes. 780 00:45:08,526 --> 00:45:10,490 AUDIENCE: So this should be a 4, 3, 2 face. 781 00:45:13,436 --> 00:45:14,418 PROFESSOR: Yep. 782 00:45:14,418 --> 00:45:15,891 AUDIENCE: But I don't know if it's the end of the 783 00:45:15,891 --> 00:45:17,141 [INAUDIBLE]. 784 00:45:21,783 --> 00:45:24,238 Could be m3m. 785 00:45:24,238 --> 00:45:25,488 AUDIENCE: Where are the-- 786 00:45:28,170 --> 00:45:29,610 it doesn't an inversion center. 787 00:45:29,610 --> 00:45:30,296 PROFESSOR: Hmm? 788 00:45:30,296 --> 00:45:32,470 AUDIENCE: It doesn't have an inversion center here. 789 00:45:32,470 --> 00:45:33,140 AUDIENCE: Oh yeah, it does. 790 00:45:33,140 --> 00:45:33,790 PROFESSOR: Oh yes, it does. 791 00:45:33,790 --> 00:45:36,060 Yes, it does. 792 00:45:36,060 --> 00:45:37,430 Actually, the thing to look for are the 793 00:45:37,430 --> 00:45:38,290 high symmetry things. 794 00:45:38,290 --> 00:45:39,350 Your eye spots them. 795 00:45:39,350 --> 00:45:41,720 That's got a fourfold axis in it. 796 00:45:41,720 --> 00:45:43,440 OK 797 00:45:43,440 --> 00:45:46,140 So are there other fourfold axes? 798 00:45:46,140 --> 00:45:47,720 Yes, yes. 799 00:45:47,720 --> 00:45:50,930 So that has to be based on 4, 3, 2. 800 00:45:50,930 --> 00:45:52,970 Because you have three orthogonal fourfold axes. 801 00:45:52,970 --> 00:45:54,710 Threefold is here. 802 00:45:54,710 --> 00:45:55,860 Twofold axis is here. 803 00:45:55,860 --> 00:45:57,110 AUDIENCE: OK. 804 00:46:06,045 --> 00:46:07,295 PROFESSOR: OK, thanks. 805 00:46:09,588 --> 00:46:11,049 AUDIENCE: There are-- 806 00:46:11,049 --> 00:46:12,742 oh, I already saw those ones. 807 00:46:12,742 --> 00:46:14,370 PROFESSOR: You want to look at some more? 808 00:46:14,370 --> 00:46:15,170 AUDIENCE: All right. 809 00:46:15,170 --> 00:46:16,420 PROFESSOR: Enough for one day? 810 00:46:21,696 --> 00:46:23,687 AUDIENCE: Professor Wuensch, I've been staring at this one 811 00:46:23,687 --> 00:46:24,735 for such a long time. 812 00:46:24,735 --> 00:46:28,400 And I couldn't match it with anything on here. 813 00:46:28,400 --> 00:46:34,090 PROFESSOR: This looks like a tetrahedron except the face is 814 00:46:34,090 --> 00:46:35,340 puckered up into this little thing. 815 00:46:35,340 --> 00:46:37,810 So the way I would start with saying, OK, this is 816 00:46:37,810 --> 00:46:41,880 tetrahedral in which case if I make these things flat, that's 817 00:46:41,880 --> 00:46:43,130 a perfect tetrahedron. 818 00:46:43,130 --> 00:46:44,500 Four sides. 819 00:46:44,500 --> 00:46:46,590 But what's happened is that this face is 820 00:46:46,590 --> 00:46:47,840 not quite 1, 1, 1. 821 00:46:50,800 --> 00:46:55,740 The 4 bar axes come out these three directions. 822 00:46:55,740 --> 00:47:00,220 And they have got twofold symmetry, mirror planes 823 00:47:00,220 --> 00:47:02,550 running through the twofold axes. 824 00:47:02,550 --> 00:47:05,940 This is the 4 bar axis with mirror planes running this 825 00:47:05,940 --> 00:47:07,490 way, this way. 826 00:47:07,490 --> 00:47:12,603 So this is actually 4 bar. 827 00:47:20,300 --> 00:47:21,840 This is still further. 828 00:47:21,840 --> 00:47:23,090 It's a cubic crystal. 829 00:47:26,850 --> 00:47:31,360 This is based on the tetrahedral symmetry. 830 00:47:31,360 --> 00:47:35,115 Here is the 4 bar axis. 831 00:47:39,530 --> 00:47:43,460 Mirror plane this way, mirror plane this way. 832 00:47:43,460 --> 00:47:47,260 Another 4 bar coming out of these two edges. 833 00:47:47,260 --> 00:47:50,020 And then these are the diagonal mirror planes. 834 00:47:50,020 --> 00:47:53,635 So this is it. 835 00:47:53,635 --> 00:47:56,373 AUDIENCE: I think I was looking for a threefold axis 836 00:47:56,373 --> 00:47:57,100 in the center. 837 00:47:57,100 --> 00:48:02,720 PROFESSOR: No, the 4 bar comes out of the edge of the 838 00:48:02,720 --> 00:48:03,970 tetrahedron. 839 00:48:05,600 --> 00:48:09,690 And this is the other 4 bar coming out here. 840 00:48:09,690 --> 00:48:13,270 Threefold like this and mirror planes going through the 841 00:48:13,270 --> 00:48:15,312 threefold axis like that. 842 00:48:15,312 --> 00:48:18,230 AUDIENCE: I see it. 843 00:48:18,230 --> 00:48:19,860 PROFESSOR: It's hard when you look at this thing. 844 00:48:19,860 --> 00:48:21,060 I never saw this thing before. 845 00:48:21,060 --> 00:48:22,020 What's here? 846 00:48:22,020 --> 00:48:25,410 But if you know the results have to be 1 of these 32 847 00:48:25,410 --> 00:48:27,580 possibilities, and this is obviously cubic, and it's 848 00:48:27,580 --> 00:48:30,580 based on the tetrahedron, that means there are only three 849 00:48:30,580 --> 00:48:32,350 possibilities. 850 00:48:32,350 --> 00:48:33,820 No mirror planes. 851 00:48:33,820 --> 00:48:37,110 If you find one mirror plane, you ask yourself, does the 852 00:48:37,110 --> 00:48:39,590 mirror plane go through the threefold axis, or does it 853 00:48:39,590 --> 00:48:41,110 miss the threefold axis? 854 00:48:41,110 --> 00:48:43,930 It goes through the threefold axis, so it's got to be this. 855 00:48:43,930 --> 00:48:47,500 And then you know just what to look for. 856 00:48:47,500 --> 00:48:51,650 AUDIENCE: Is it just [INAUDIBLE] m? 857 00:48:51,650 --> 00:48:53,120 PROFESSOR: This is a dirty one. 858 00:48:53,120 --> 00:48:54,130 This is a dirty one. 859 00:48:54,130 --> 00:48:58,140 If you look at this very carefully, this face is the 860 00:48:58,140 --> 00:49:01,265 same as this face, but it's not the same as that one. 861 00:49:01,265 --> 00:49:02,850 It's a little bigger. 862 00:49:02,850 --> 00:49:07,760 So if you say, what's in here, these two things do not come 863 00:49:07,760 --> 00:49:11,400 together at a common vertex like these two. 864 00:49:11,400 --> 00:49:15,250 There's another little line segment between these faces. 865 00:49:15,250 --> 00:49:18,380 So there are no threefold axes coming out of this. 866 00:49:18,380 --> 00:49:20,230 Looks like it might be tetrahedral. 867 00:49:20,230 --> 00:49:23,070 Clearly, a mirror plane going this way. 868 00:49:23,070 --> 00:49:30,180 And then there is also a twofold axis coming out here. 869 00:49:30,180 --> 00:49:37,430 So there's a 2 over m and another 2 over m. 870 00:49:47,440 --> 00:49:50,890 I did this once before with somebody, and this is the 871 00:49:50,890 --> 00:49:52,678 hardest one in the whole set. 872 00:50:02,305 --> 00:50:03,780 So a mirror plane this way. 873 00:50:06,710 --> 00:50:08,570 There's got to be another mirror plane because there's a 874 00:50:08,570 --> 00:50:09,820 twofold axis. 875 00:50:17,440 --> 00:50:19,162 4 bar. 876 00:50:19,162 --> 00:50:24,940 Up, down, up, down. 877 00:50:24,940 --> 00:50:31,878 So this is 4 bar, 2 over m. 878 00:50:34,560 --> 00:50:35,810 4 bar, 2 over m.