WEBVTT 00:00:07.110 --> 00:00:09.890 PROFESSOR: --questions about what we've done. 00:00:09.890 --> 00:00:13.460 I think it's been fast, but hopefully if you understand 00:00:13.460 --> 00:00:16.500 the principles, we didn't go too fast. 00:00:16.500 --> 00:00:19.190 But any questions on what we've done? 00:00:24.470 --> 00:00:25.720 I guess you haven't had a chance to think 00:00:25.720 --> 00:00:27.980 of questions yet. 00:00:27.980 --> 00:00:33.240 Let me take care of our oddball symmetry at the end of 00:00:33.240 --> 00:00:38.150 the chart that I handed out-- and this 4 bar-- 00:00:38.150 --> 00:00:42.110 and ask what we can do there. 00:00:42.110 --> 00:00:50.060 Having discovered the four bar operation in 2, 2, 2 with 00:00:50.060 --> 00:00:53.960 diagonal mirror planes, we can consider that as a new type of 00:00:53.960 --> 00:00:55.650 symmetry element. 00:00:55.650 --> 00:01:02.450 And this is the symbol for it. 00:01:02.450 --> 00:01:07.430 And this would take a pair of objects and repeat them by a 00:01:07.430 --> 00:01:09.440 180 degree rotation. 00:01:09.440 --> 00:01:13.640 And then another pair of opposite handedness, opposite 00:01:13.640 --> 00:01:18.980 chirality would be rotated 90 degrees and inverted, rotated 00:01:18.980 --> 00:01:21.210 90 degrees and inverted. 00:01:21.210 --> 00:01:29.110 And I mentioned last time that a solid that has this shape is 00:01:29.110 --> 00:01:31.130 something called a sphenoid. 00:01:34.700 --> 00:01:39.580 And the 4 bar axis takes a pair of faces that are up and 00:01:39.580 --> 00:01:43.940 a pair of faces that are down and skewed by 90 degrees. 00:01:43.940 --> 00:01:45.890 Now what can we do? 00:01:45.890 --> 00:01:52.780 If we add a mirror plane that's perpendicular to the 4 00:01:52.780 --> 00:01:55.850 bar axis, the right-handed one goes down, the 00:01:55.850 --> 00:01:57.480 left-handed one comes up. 00:01:57.480 --> 00:02:03.200 And it becomes simply 4 over m, which we've already got. 00:02:03.200 --> 00:02:08.725 If we add a vertical mirror plane through the 4 bar, if 00:02:08.725 --> 00:02:18.040 you look a bit earlier at 4 bar 2m, now we've got the 4 00:02:18.040 --> 00:02:21.420 bar, now we've got the mirror plane. 00:02:21.420 --> 00:02:26.070 Not surprisingly, we get the two mirror planes. 00:02:26.070 --> 00:02:41.200 And so S4v is going to be the same as 4 bar 2m. 00:02:44.380 --> 00:02:48.400 And that's something we already have D2d. 00:02:48.400 --> 00:02:50.030 So that exhausts the possibilities. 00:02:50.030 --> 00:02:51.595 4 bar just stands by itself. 00:02:55.220 --> 00:02:58.790 At a vertical mirror plane, it's D2d 4 bar 2m. 00:02:58.790 --> 00:03:01.920 At a horizontal mirror plane, it becomes 4 over m. 00:03:01.920 --> 00:03:08.300 The diagonal mirror plane is not possible. 00:03:08.300 --> 00:03:11.500 There's nothing to place the mirror plane diagonal to. 00:03:11.500 --> 00:03:14.290 And if you add inversion, it changes into 4 over m. 00:03:14.290 --> 00:03:16.100 So this one is an odd group. 00:03:16.100 --> 00:03:19.330 It sits by itself. 00:03:19.330 --> 00:03:21.860 Except that we could start there, and add a vertical 00:03:21.860 --> 00:03:27.070 mirror plane, and get 4 bar 2m once more. 00:03:27.070 --> 00:03:31.160 That leaves the cubic ones, which are really easy to deal 00:03:31.160 --> 00:03:33.490 with because a cube has such high symmetry. 00:03:33.490 --> 00:03:35.450 We all know what cubes look like. 00:03:35.450 --> 00:03:39.210 But when you show arrangements of motifs, it gets a little 00:03:39.210 --> 00:03:41.070 bit confusing. 00:03:41.070 --> 00:03:44.295 So let's take a look at the tetrahedral groups. 00:03:47.630 --> 00:03:53.360 T is a combination of twofold axes coming out in directions 00:03:53.360 --> 00:03:55.670 corresponding to face normals to a cube. 00:04:00.590 --> 00:04:02.865 So these are the twofold axes. 00:04:08.510 --> 00:04:14.140 And then there are threefold axes coming out of directions 00:04:14.140 --> 00:04:17.760 that correspond to the face diagonal. 00:04:17.760 --> 00:04:22.060 So this is the jack-o'-lantern stereographic projection, 00:04:22.060 --> 00:04:27.410 which is something I love to draw at this time of year. 00:04:27.410 --> 00:04:29.360 What will the pattern look like? 00:04:29.360 --> 00:04:33.760 Well, it's going to look like 2, 2, 2. 00:04:37.010 --> 00:04:38.200 So that's a subgroup. 00:04:38.200 --> 00:04:42.050 So let's draw in a pair of objects on either side of the 00:04:42.050 --> 00:04:45.774 twofold axis, and that's the pattern of 2, 2, 2. 00:04:45.774 --> 00:04:50.420 Now this guy here is lurking off of a threefold axis. 00:04:50.420 --> 00:04:54.570 So that threefold axis is going to repeat it three 00:04:54.570 --> 00:04:58.800 locations that are 120 degrees apart. 00:04:58.800 --> 00:05:03.700 And so there's going to be a triangle of objects up above. 00:05:03.700 --> 00:05:10.660 This threefold axis is going to reproduce this into a 00:05:10.660 --> 00:05:13.660 triangle of objects that are down below. 00:05:13.660 --> 00:05:17.100 The twofold axis will take this triangle of objects and 00:05:17.100 --> 00:05:19.390 move it over to here. 00:05:19.390 --> 00:05:21.090 Also, up. 00:05:21.090 --> 00:05:24.110 And the twofold axis that's vertical will take this 00:05:24.110 --> 00:05:29.450 triangle and move it down to three that are below. 00:05:29.450 --> 00:05:31.540 So it looks very complicated in projection. 00:05:31.540 --> 00:05:35.700 But it's simply a planar triangle of atoms here rotated 00:05:35.700 --> 00:05:36.740 180 degrees. 00:05:36.740 --> 00:05:38.330 So you've got one like this, one like this. 00:05:38.330 --> 00:05:41.925 And then down below, two other planar triangles of objects. 00:05:47.700 --> 00:05:58.050 Remembering what the symmetry of a tetrahedron looks like, 00:05:58.050 --> 00:06:04.150 you can draw this arrangement of symmetry elements relative 00:06:04.150 --> 00:06:06.510 to a tetrahedron. 00:06:06.510 --> 00:06:09.890 And the threefold axes now are coming out 00:06:09.890 --> 00:06:11.365 normal to the faces. 00:06:11.365 --> 00:06:14.160 The twofold axes coming out normal to the edges. 00:06:16.810 --> 00:06:20.800 And imagine that we put a triangle on this face, a 00:06:20.800 --> 00:06:24.700 similar triangle on the face behind, and a triangle 00:06:24.700 --> 00:06:27.300 pointing in the other direction down this way. 00:06:27.300 --> 00:06:28.830 These are the three below. 00:06:28.830 --> 00:06:31.960 So that's the pattern of 2, 3. 00:06:31.960 --> 00:06:35.400 Take a tetrahedron, smack a triangle on the two upper 00:06:35.400 --> 00:06:39.130 faces, and an equilateral triangle on 00:06:39.130 --> 00:06:41.590 the two lower faces. 00:06:41.590 --> 00:06:44.880 And the threefold axis that comes out here comes out of 00:06:44.880 --> 00:06:47.000 the center of this triangle and out of the 00:06:47.000 --> 00:06:48.250 center of this triangle. 00:06:59.910 --> 00:07:06.230 The next one that you can get from this is Th, the 00:07:06.230 --> 00:07:08.530 tetrahedral arrangement of axes with a 00:07:08.530 --> 00:07:11.470 horizontal mirror plane. 00:07:11.470 --> 00:07:14.290 If you put in a horizontal mirror plane, it's going to 00:07:14.290 --> 00:07:19.850 take this triangle and reflect it down. 00:07:19.850 --> 00:07:22.540 It's going to take this triangle and reflect it up. 00:07:22.540 --> 00:07:26.336 And they will overlap in projection. 00:07:26.336 --> 00:07:30.440 And so that's the pattern for Th. 00:07:30.440 --> 00:07:32.470 There's a mirror plane perpendicular 00:07:32.470 --> 00:07:34.880 to a twofold axis. 00:07:34.880 --> 00:07:36.920 That creates an inversion center. 00:07:36.920 --> 00:07:39.200 So we can call this a 3 bar axis. 00:07:39.200 --> 00:07:44.360 So that's Th 2 over m 3 bar. 00:07:50.720 --> 00:07:56.715 So just have triangles on the upper and lower faces as well. 00:08:03.270 --> 00:08:11.970 The one remaining one is Td. 00:08:17.760 --> 00:08:19.850 Here are the twofold axes. 00:08:19.850 --> 00:08:24.620 Here are the threefold axes. 00:08:24.620 --> 00:08:28.780 And now if we put a diagonal mirror plane in, diagonal with 00:08:28.780 --> 00:08:29.690 respect to what? 00:08:29.690 --> 00:08:32.390 Well, diagonal with respect to these twofold axes. 00:08:32.390 --> 00:08:37.140 So this mirror plane goes down like this, passes through a 00:08:37.140 --> 00:08:38.110 twofold axis. 00:08:38.110 --> 00:08:42.799 There must be a mirror plane 90 degrees away. 00:08:42.799 --> 00:08:50.170 And the threefold axis is going to repeat these mirror 00:08:50.170 --> 00:08:58.270 planes so that we get mirror planes that are at an angle 00:08:58.270 --> 00:09:01.210 with respect to the vertical twofold axis. 00:09:01.210 --> 00:09:05.100 And this is Td. 00:09:05.100 --> 00:09:10.800 And this, in the international notation, is called-- if we 00:09:10.800 --> 00:09:12.320 can figure it out-- 00:09:12.320 --> 00:09:16.470 the diagonal mirror planes with respect to these twofold 00:09:16.470 --> 00:09:20.250 axes have changed them into 4 bar axes. 00:09:20.250 --> 00:09:26.740 So this is called 4 bar 3m, which doesn't 00:09:26.740 --> 00:09:28.230 look cubic at all. 00:09:28.230 --> 00:09:29.555 So that's a little bit deceptive. 00:09:36.970 --> 00:09:41.960 I am foolhardy for even trying to do this. 00:09:41.960 --> 00:09:44.280 And so I don't think I'm going to do it. 00:09:44.280 --> 00:09:50.930 4, 3, 2, we know what that looks like. 00:09:50.930 --> 00:09:53.930 That is fourfold axes coming out in directions that 00:09:53.930 --> 00:10:01.115 correspond to the face normals to a cube. 00:10:01.115 --> 00:10:06.110 A threefold axis coming out of body diagonals. 00:10:06.110 --> 00:10:09.460 Twofold axes between all of the fourfold axes that are 00:10:09.460 --> 00:10:11.170 normal to the edges of the cube. 00:10:13.910 --> 00:10:17.830 So all sorts of rotational symmetry. 00:10:17.830 --> 00:10:24.020 That, if you want a pattern, has a triangle that's on about 00:10:24.020 --> 00:10:28.600 each of the threefold axes. 00:10:28.600 --> 00:10:33.970 But the one that is up, points in the opposite orientation of 00:10:33.970 --> 00:10:37.760 the one that's on the threefold axis coming out of 00:10:37.760 --> 00:10:38.860 the other end of the cube. 00:10:38.860 --> 00:10:42.280 So you have one triangle that's like this, up. 00:10:42.280 --> 00:10:45.810 And another triangle like this that's on the other end of the 00:10:45.810 --> 00:10:48.570 threefold axis that points down into the blackboard. 00:10:53.630 --> 00:10:57.440 What can you add as other extenders? 00:10:57.440 --> 00:11:00.890 You can put in a mirror plane that is perpendicular to the 00:11:00.890 --> 00:11:02.480 fourfold axis. 00:11:02.480 --> 00:11:04.973 And that leaves everything unchanged. 00:11:09.740 --> 00:11:12.720 Now we've got a mirror plane passing through a fourfold 00:11:12.720 --> 00:11:15.370 axis, so we have to have mirror planes 00:11:15.370 --> 00:11:18.420 at 45 degree intervals. 00:11:18.420 --> 00:11:21.650 And that will create mirror planes there. 00:11:21.650 --> 00:11:24.050 This horizontal mirror plane goes through this fourfold 00:11:24.050 --> 00:11:28.370 axis, so there must be mirror planes at 45 degree intervals. 00:11:28.370 --> 00:11:30.040 So there's another one like this and 90 00:11:30.040 --> 00:11:31.930 degrees away as well. 00:11:31.930 --> 00:11:34.680 The fourfold axis is perpendicular to a mirror 00:11:34.680 --> 00:11:37.650 plane, as are these other twofold axes, so there's an 00:11:37.650 --> 00:11:39.140 inversion center. 00:11:39.140 --> 00:11:42.600 Here's a fourfold axis with a vertical mirror plane going 00:11:42.600 --> 00:11:42.930 through it. 00:11:42.930 --> 00:11:45.885 And it has to have a vertical mirror plane going like this 00:11:45.885 --> 00:11:47.090 at 45 degrees away. 00:11:47.090 --> 00:11:48.400 So that's the symmetry. 00:11:48.400 --> 00:11:52.910 This is the regular symmetry of a cube or an octahedron. 00:11:52.910 --> 00:11:55.880 And I will not, even if pressured, try to draw a 00:11:55.880 --> 00:11:59.870 pattern that conforms to that. 00:11:59.870 --> 00:12:01.990 We've got the fourfold axis 00:12:01.990 --> 00:12:04.990 perpendicular to a mirror plane. 00:12:04.990 --> 00:12:07.820 And this is called O sub h, O with a 00:12:07.820 --> 00:12:11.030 horizontal mirror plane. 00:12:11.030 --> 00:12:17.590 The fourfold axis in 4, 3, 2 has got a mirror plane 00:12:17.590 --> 00:12:18.670 perpendicular to it. 00:12:18.670 --> 00:12:20.860 The twofold axes all pick up mirror planes 00:12:20.860 --> 00:12:22.230 perpendicular to them. 00:12:22.230 --> 00:12:25.270 There's an inversion center at the point of intersection. 00:12:25.270 --> 00:12:28.520 So we label this axis a 3 bar axis. 00:12:28.520 --> 00:12:32.840 So that's O sub h, 4 over m, 3 bar, 2 over m. 00:12:32.840 --> 00:12:37.655 And this, as I say, is the symmetry of a regular cube or 00:12:37.655 --> 00:12:40.280 an octahedron. 00:12:40.280 --> 00:12:41.580 Nothing else we can do here. 00:12:41.580 --> 00:12:44.730 There's so many symmetry axes all over the place that 00:12:44.730 --> 00:12:48.730 there's no way we can snake in any diagonal mirror plane. 00:12:48.730 --> 00:12:53.070 Because there is no second axis of the same kind adjacent 00:12:53.070 --> 00:12:55.360 to either the twofold, the threefold, or 00:12:55.360 --> 00:12:56.400 the fourfold axis. 00:12:56.400 --> 00:12:59.230 So we can't put a mirror plane in here between the threefold 00:12:59.230 --> 00:13:00.680 and the twofold. 00:13:00.680 --> 00:13:03.245 We can't put a mirror plane in here between the fourfold and 00:13:03.245 --> 00:13:04.130 the twofold. 00:13:04.130 --> 00:13:05.060 So we're done. 00:13:05.060 --> 00:13:09.250 And this is the final and 30 second combination. 00:13:09.250 --> 00:13:14.082 So there are 32 crystallographic point groups. 00:13:14.082 --> 00:13:14.570 AUDIENCE: Professor? 00:13:14.570 --> 00:13:15.280 PROFESSOR: Yes, sir. 00:13:15.280 --> 00:13:22.084 AUDIENCE: So if m3m is a regular operation, are there 00:13:22.084 --> 00:13:23.334 symbols [INAUDIBLE]? 00:13:26.960 --> 00:13:29.190 PROFESSOR: O is the symbol for 4, 3, 2. 00:13:29.190 --> 00:13:32.690 And what we added as an extender is a mirror plane 00:13:32.690 --> 00:13:34.550 perpendicular to the fourfold axis. 00:13:34.550 --> 00:13:38.000 That is just one way of adding an extender. 00:13:38.000 --> 00:13:39.800 This is also Oi. 00:13:39.800 --> 00:13:42.690 If we add an inversion center, we get all this. 00:13:42.690 --> 00:13:46.090 And if you look at the ones have been honored by being 00:13:46.090 --> 00:13:51.250 designated by a symbol, the horizontal mirror plane takes 00:13:51.250 --> 00:13:53.230 the precedence. 00:13:53.230 --> 00:13:57.510 So for example, for 2 over m, 2 over m, 2 over m, you've got 00:13:57.510 --> 00:13:59.930 two different vertical mirror planes, and you've got one 00:13:59.930 --> 00:14:01.360 horizontal mirror plane. 00:14:01.360 --> 00:14:04.820 But it's called 2 over m, 2 over m, 2 over m. 00:14:04.820 --> 00:14:07.670 You can call it 2, 2, 2, m, m, m. 00:14:07.670 --> 00:14:12.050 That's also a possible symbol, but much more of a mouthful. 00:14:12.050 --> 00:14:15.720 There's some arbitrariness to the symbol because the 00:14:15.720 --> 00:14:18.900 arrangement of symmetry elements is what's real. 00:14:18.900 --> 00:14:22.180 And we decide how we want to devise a 00:14:22.180 --> 00:14:24.730 notation to label them. 00:14:24.730 --> 00:14:27.660 And as we've seen, there are two different people who 00:14:27.660 --> 00:14:30.690 adopted a different code for giving them names. 00:14:30.690 --> 00:14:32.870 So if it's rational and 00:14:32.870 --> 00:14:38.610 informative, it's a good notation. 00:14:38.610 --> 00:14:41.650 Interestingly, the international notation, on the 00:14:41.650 --> 00:14:44.520 one hand, and the Schoenflies notation, on the other, 00:14:44.520 --> 00:14:46.090 complimentary. 00:14:46.090 --> 00:14:48.280 The international notation tells you 00:14:48.280 --> 00:14:51.260 unambiguously what you have. 00:14:51.260 --> 00:14:56.310 So 2, 2, 2 over m, m, m tells you three orthogonal twofold 00:14:56.310 --> 00:14:58.400 axes if you remember what came out of [INAUDIBLE] 00:14:58.400 --> 00:14:59.320 construction. 00:14:59.320 --> 00:15:02.480 And perpendicular to each of them is a mirror plane. 00:15:02.480 --> 00:15:04.170 And that's what you've got. 00:15:04.170 --> 00:15:07.590 Schoenflies tells you how you derived it. 00:15:07.590 --> 00:15:13.120 You took D2, and that's the dihedral group 2, 2, 2, and 00:15:13.120 --> 00:15:16.340 you added an h, a horizontal mirror plane. 00:15:16.340 --> 00:15:17.490 And all hell broke loose. 00:15:17.490 --> 00:15:19.130 And you got mirror planes perpendicular to all the 00:15:19.130 --> 00:15:20.320 twofold axes. 00:15:20.320 --> 00:15:22.950 So there's a certain complementarity to the two 00:15:22.950 --> 00:15:24.950 different notations. 00:15:24.950 --> 00:15:29.870 And the people who do diffraction and 00:15:29.870 --> 00:15:34.760 crystallography for the most part follow the international 00:15:34.760 --> 00:15:37.520 notation, the Hermann-Mauguin notation. 00:15:37.520 --> 00:15:40.810 The people who do condensed matter physics use the 00:15:40.810 --> 00:15:42.340 Schoenflies notation. 00:15:42.340 --> 00:15:45.490 Because it's more inscrutable. 00:15:45.490 --> 00:15:47.670 And condensed matter physicists like to be 00:15:47.670 --> 00:15:52.190 inscrutable because it's how you gain respect. 00:15:52.190 --> 00:15:58.080 So both notation survive, but are more prevalent in some 00:15:58.080 --> 00:16:00.930 disciplines than in others and vice versa. 00:16:08.080 --> 00:16:11.590 Let's take a brief look ahead. 00:16:11.590 --> 00:16:13.610 What remains to be done? 00:16:13.610 --> 00:16:18.465 We now have the 32 crystallographic point groups. 00:16:21.280 --> 00:16:25.410 These are the way things can be arranged by symmetry about 00:16:25.410 --> 00:16:28.420 a fixed point in space. 00:16:28.420 --> 00:16:31.230 If we were to proceed in the same way that we did for the 00:16:31.230 --> 00:16:34.620 two-dimensional space groups, what we should do next is 00:16:34.620 --> 00:16:38.960 decide what sort of three-dimensional space 00:16:38.960 --> 00:16:42.760 lattices these symmetries will require. 00:16:42.760 --> 00:16:49.110 And then proceed to drop each of the point groups into each 00:16:49.110 --> 00:16:52.660 of the lattices that can accommodate them. 00:16:52.660 --> 00:16:56.060 And then use the tricks that we've used in two dimensions, 00:16:56.060 --> 00:16:59.920 take mirror planes and replace them by glide planes. 00:16:59.920 --> 00:17:03.130 And then having done that, we would take the mirror planes 00:17:03.130 --> 00:17:08.160 and the rotation axes and interweave them. 00:17:08.160 --> 00:17:12.060 And lest that job seem too daunting, let me point out 00:17:12.060 --> 00:17:15.760 that we already have 17 of the space groups. 00:17:15.760 --> 00:17:19.089 All we have to do is take the plane groups, take a 00:17:19.089 --> 00:17:21.579 translation that's perpendicular to the plane of 00:17:21.579 --> 00:17:24.914 the plane group, and let all the rotation axes and mirror 00:17:24.914 --> 00:17:28.054 planes extend up indefinitely in three dimensions parallel 00:17:28.054 --> 00:17:30.600 to that translation. 00:17:30.600 --> 00:17:33.740 So we already have 17 of the 00:17:33.740 --> 00:17:35.360 three-dimensional space groups. 00:17:35.360 --> 00:17:39.430 They look just like the plane groups except they extend in a 00:17:39.430 --> 00:17:43.020 direction that's perpendicular to the plane of our original 00:17:43.020 --> 00:17:45.030 two-dimensional group. 00:17:45.030 --> 00:17:48.260 So we're already a long way towards deriving a 00:17:48.260 --> 00:17:52.020 three-dimensional space group without really knowing it. 00:17:52.020 --> 00:17:55.130 But what I would like to consider next is the lattices 00:17:55.130 --> 00:17:57.870 that are required for three dimensions. 00:17:57.870 --> 00:18:02.160 And I've already given you how one can approach this problem. 00:18:02.160 --> 00:18:07.360 Take the lattices that have been required in the 00:18:07.360 --> 00:18:09.190 two-dimensional plane groups. 00:18:09.190 --> 00:18:13.280 And if the presence of a threefold axis and the base of 00:18:13.280 --> 00:18:19.510 the cell require, say, net that is hexagonal with a1 00:18:19.510 --> 00:18:29.160 equal to a2 identically in magnitude and exactly at 120 00:18:29.160 --> 00:18:30.945 degrees with respect to one another. 00:18:35.300 --> 00:18:39.770 Let's let the third translation be perpendicular 00:18:39.770 --> 00:18:42.170 to the base. 00:18:42.170 --> 00:18:44.430 If it's hexagonal, we have to have at least a 00:18:44.430 --> 00:18:47.690 threefold axis here. 00:18:47.690 --> 00:18:51.270 And we've found that adding a threefold axis to that net 00:18:51.270 --> 00:18:53.830 gave rise to two other threefold axes in the center 00:18:53.830 --> 00:18:55.200 of the triangle. 00:18:55.200 --> 00:18:59.740 So one lattice would be one in which the third translation, 00:18:59.740 --> 00:19:03.850 which we'll label c, is straight up. 00:19:08.530 --> 00:19:11.670 In other words, perpendicular to the net that we derived in 00:19:11.670 --> 00:19:14.010 two dimensions. 00:19:14.010 --> 00:19:17.320 What we will have as a constraint is that if we pick 00:19:17.320 --> 00:19:21.180 a certain translation that is not normal to the plane group, 00:19:21.180 --> 00:19:25.080 and let's say it terminates here in projection. 00:19:25.080 --> 00:19:28.130 So the third translation goes up and over. 00:19:28.130 --> 00:19:31.380 If there's a threefold axis at this end of the translation, 00:19:31.380 --> 00:19:34.010 there must be a threefold axis at the end of that 00:19:34.010 --> 00:19:35.280 translation. 00:19:35.280 --> 00:19:39.200 And that mucks everything up, and we no longer have a group. 00:19:39.200 --> 00:19:41.880 We can't have another threefold axis poking down 00:19:41.880 --> 00:19:45.520 through the plane of this two-dimensional plane group. 00:19:45.520 --> 00:19:50.300 But it is perfectly OK if the third translation moves this 00:19:50.300 --> 00:19:53.830 threefold axis and puts down a lattice point and another 00:19:53.830 --> 00:19:57.590 threefold axis directly over this one. 00:19:57.590 --> 00:20:00.270 Or alternatively, another choice for T3 00:20:00.270 --> 00:20:01.580 would be this one. 00:20:01.580 --> 00:20:04.980 That would put the lattice point in a threefold axis 00:20:04.980 --> 00:20:06.120 directly over this one. 00:20:06.120 --> 00:20:09.790 So again, we have threefold axes extending normal to the 00:20:09.790 --> 00:20:11.123 plane of my drawing. 00:20:11.123 --> 00:20:14.370 And I've created no threefold axes. 00:20:14.370 --> 00:20:19.220 So with a threefold axis, there are three potentially 00:20:19.220 --> 00:20:23.770 different space lattices, one in which the third translation 00:20:23.770 --> 00:20:26.760 is normal to the plane of the plane group, another one where 00:20:26.760 --> 00:20:28.900 the translation terminates over the 00:20:28.900 --> 00:20:31.200 other two twofold axes. 00:20:31.200 --> 00:20:33.290 And so this is the way in which one would proceed to 00:20:33.290 --> 00:20:36.950 derive the space lattices. 00:20:36.950 --> 00:20:38.710 Buerger does not do it this way. 00:20:38.710 --> 00:20:40.180 He doesn't look at the plane groups. 00:20:40.180 --> 00:20:45.150 He looks just at where the axes sit in the nets. 00:20:45.150 --> 00:20:47.150 No mirror planes whatsoever. 00:20:47.150 --> 00:20:48.610 And I'll return to the point. 00:20:48.610 --> 00:20:52.490 And the place where I think he's wrong is that if you add 00:20:52.490 --> 00:21:00.410 a twofold axis to a centered net, you get twofold axes in 00:21:00.410 --> 00:21:04.131 all of these locations. 00:21:04.131 --> 00:21:06.620 These are the twofold axes in C2mm. 00:21:10.000 --> 00:21:13.970 And if that's all you look at, there's no reason why this 00:21:13.970 --> 00:21:17.100 should be the plane group. 00:21:17.100 --> 00:21:20.800 And it looks as though you can make T3 terminate over this 00:21:20.800 --> 00:21:22.220 twofold axis. 00:21:22.220 --> 00:21:25.710 And that would give you a peculiar triple cell with 00:21:25.710 --> 00:21:28.680 three lattice points along the long diagonal of 00:21:28.680 --> 00:21:31.720 a rectangular net. 00:21:31.720 --> 00:21:35.490 And you say, wow, 15 space lattices. 00:21:35.490 --> 00:21:38.970 We're going to be famous, if not rich. 00:21:38.970 --> 00:21:43.620 You can't do that because this plane group can exist only if 00:21:43.620 --> 00:21:46.330 there are mirror planes in here like this. 00:21:46.330 --> 00:21:50.590 And then through these twofold axes, these have to be glides. 00:21:50.590 --> 00:21:54.820 So you can't take 2mm and put it on top of 2gg. 00:21:54.820 --> 00:21:56.830 It's impossible. 00:21:56.830 --> 00:22:00.050 So there are 14 space lattices. 00:22:00.050 --> 00:22:01.500 The 15th doesn't exist. 00:22:01.500 --> 00:22:04.000 But it looks as though it's possible in Buerger's 00:22:04.000 --> 00:22:06.040 treatment when he looks just at the placement of the 00:22:06.040 --> 00:22:09.180 rotation axes alone and not the location of any mirror 00:22:09.180 --> 00:22:13.040 planes that are in the plane groups. 00:22:13.040 --> 00:22:16.010 So that is where we're going to go next. 00:22:16.010 --> 00:22:19.000 And that is a process that actually is 00:22:19.000 --> 00:22:21.670 surprisingly simple. 00:22:21.670 --> 00:22:25.120 And we will get the space lattices very quickly. 00:22:25.120 --> 00:22:27.430 We'll get a number of space groups along the way, 00:22:27.430 --> 00:22:28.740 as we've just seen. 00:22:28.740 --> 00:22:32.100 And then we will cut to the bottom line and just look at 00:22:32.100 --> 00:22:36.700 how this information is tabulated in tables for you in 00:22:36.700 --> 00:22:39.230 a fashion that's analogous to the representation of the 00:22:39.230 --> 00:22:40.420 plane groups. 00:22:40.420 --> 00:22:41.260 So we're pretty close. 00:22:41.260 --> 00:22:44.325 We'll be wrapping things up in another two or three meetings. 00:22:46.830 --> 00:22:50.590 What I would like to do in the time that remains though, the 00:22:50.590 --> 00:22:54.496 point groups are very difficult to visualize unless 00:22:54.496 --> 00:22:59.130 you look at real crystals. 00:22:59.130 --> 00:23:03.640 So what I'm going to do is pass around a collection of 00:23:03.640 --> 00:23:05.360 models of actual crystals. 00:23:05.360 --> 00:23:09.870 I'm not sure I can tell you what every one of these models 00:23:09.870 --> 00:23:11.120 represents. 00:23:11.120 --> 00:23:12.740 These are curious things. 00:23:12.740 --> 00:23:15.980 I would beg you not to drop them on one 00:23:15.980 --> 00:23:19.030 of the sharp corners. 00:23:19.030 --> 00:23:20.210 These are made out of wood. 00:23:20.210 --> 00:23:25.200 They're made out of pear wood in the Black Forest by elves. 00:23:25.200 --> 00:23:28.230 No, that last part is not true. 00:23:28.230 --> 00:23:30.700 But they are incredibly expensive. 00:23:30.700 --> 00:23:32.880 Because the angles in these things have 00:23:32.880 --> 00:23:34.780 to be exactly right. 00:23:34.780 --> 00:23:38.140 So they are made on the same sort of machine that you use 00:23:38.140 --> 00:23:40.480 for faceting diamonds. 00:23:40.480 --> 00:23:43.380 It's something that has two degrees of freedom, so you can 00:23:43.380 --> 00:23:47.220 get a surface that you want on the material exactly parallel 00:23:47.220 --> 00:23:48.690 to a grinding surface. 00:23:48.690 --> 00:23:52.150 And then you have a provision for advancing it normal to 00:23:52.150 --> 00:23:54.010 that direction by a controlled amount. 00:23:54.010 --> 00:23:57.130 If it were not precise, you would not see sharp edges and 00:23:57.130 --> 00:23:58.200 sharp corners. 00:23:58.200 --> 00:24:02.000 So these are incredibly expensive. 00:24:02.000 --> 00:24:05.310 I, therefore, ask it that you hold them 00:24:05.310 --> 00:24:06.625 tightly with both hands. 00:24:06.625 --> 00:24:08.170 Don't drop them on the floor. 00:24:08.170 --> 00:24:10.420 And I'll take around a couple of handfuls of these. 00:24:10.420 --> 00:24:12.660 I don't know if I have enough for everyone. 00:24:12.660 --> 00:24:15.570 With the tables of the point groups in front of you, why 00:24:15.570 --> 00:24:19.040 don't you try to identify the point group that's possible in 00:24:19.040 --> 00:24:20.290 each of these models. 00:24:26.820 --> 00:24:27.640 This is a big one. 00:24:27.640 --> 00:24:29.487 So I'm sure you won't drop that one. 00:24:45.400 --> 00:24:47.770 I don't think I have enough for everyone, so I'll start 00:24:47.770 --> 00:24:49.530 passing them out to every other person. 00:24:49.530 --> 00:24:51.820 Maybe you can look on with the person adjacent to you. 00:25:00.269 --> 00:25:01.519 Take one of these. 00:25:18.658 --> 00:25:19.908 Pass those over. 00:25:28.598 --> 00:25:29.848 Here you go. 00:25:43.600 --> 00:25:45.970 I gave you that just to be mean. 00:25:45.970 --> 00:25:48.167 That is an example of something that's called a 00:25:48.167 --> 00:25:50.010 twinned crystal. 00:25:50.010 --> 00:25:51.770 You can't have reentrant faces. 00:25:51.770 --> 00:25:54.140 Yeah, that's actually two crystals that are intergrown. 00:25:58.880 --> 00:26:00.840 AUDIENCE: So we can't identify the [INAUDIBLE]? 00:26:00.840 --> 00:26:02.090 PROFESSOR: You can look. 00:26:02.090 --> 00:26:05.470 Block one of them out in your mind, and try to imagine what 00:26:05.470 --> 00:26:06.310 it would look like. 00:26:06.310 --> 00:26:06.580 AUDIENCE: [INAUDIBLE]. 00:26:06.580 --> 00:26:07.830 PROFESSOR: Yeah. 00:26:16.090 --> 00:26:17.990 Here's an example of another one. 00:26:17.990 --> 00:26:20.220 That's actually gypsum. 00:26:20.220 --> 00:26:23.110 And that's a very common twin in gypsum. 00:26:23.110 --> 00:26:26.580 And if you could take this and rotate it-- 00:26:26.580 --> 00:26:27.810 this one doesn't rotate-- 00:26:27.810 --> 00:26:31.660 rotate it 90 degrees, then you would have a crystal that had 00:26:31.660 --> 00:26:34.382 a parallelogram shape. 00:26:34.382 --> 00:26:37.490 And actually that's been rotated by 90 00:26:37.490 --> 00:26:39.790 degrees about an axis. 00:26:39.790 --> 00:26:41.040 That's within the base. 00:26:45.635 --> 00:26:50.366 AUDIENCE: No, no, I think that's just 4m or 4 over m. 00:26:50.366 --> 00:26:52.060 AUDIENCE: Yeah, there's no mirror there. 00:26:52.060 --> 00:26:54.720 That's a 4 bar, I think. 00:26:54.720 --> 00:26:56.570 AUDIENCE: Oh, is that the inversion? 00:26:56.570 --> 00:26:59.940 PROFESSOR: There's no-- neither. 00:26:59.940 --> 00:27:01.330 I am a touchy-feely guy. 00:27:01.330 --> 00:27:04.000 If I see something up here, I look down here and try to feel 00:27:04.000 --> 00:27:06.020 something that's parallel to it. 00:27:06.020 --> 00:27:07.350 But you're right, you're right. 00:27:07.350 --> 00:27:09.520 If I rotate 90 degrees-- 00:27:09.520 --> 00:27:12.650 remember rotoinversion is the same as rotoreflection. 00:27:12.650 --> 00:27:15.730 So I could rotate to here and then reflect down, 00:27:15.730 --> 00:27:16.980 and I get this one. 00:27:19.320 --> 00:27:20.210 AUDIENCE: So it's 4 bar. 00:27:20.210 --> 00:27:22.080 PROFESSOR: It's 4 bar, yeah. 00:27:22.080 --> 00:27:24.470 And you see in the cross section, it is square. 00:27:24.470 --> 00:27:28.260 There's a little square on top here, so there is a fourfold 00:27:28.260 --> 00:27:30.440 aspect to it when you look at planes in a special 00:27:30.440 --> 00:27:30.915 orientation. 00:27:30.915 --> 00:27:33.290 AUDIENCE: I have a question about 4 bar. 00:27:33.290 --> 00:27:38.050 4 bar in planes are twofold rotation. 00:27:38.050 --> 00:27:39.870 PROFESSOR: A twofold pure rotation. 00:27:39.870 --> 00:27:43.520 Probably the best example of 4 bar is a tetrahedron. 00:27:43.520 --> 00:27:47.350 Two faces on top, two faces twisted 90 degrees, and 00:27:47.350 --> 00:27:48.110 inverted down. 00:27:48.110 --> 00:27:50.006 AUDIENCE: So that means an 8 bar would be a 00:27:50.006 --> 00:27:50.954 fourfold proper rotation. 00:27:50.954 --> 00:27:53.798 So why cannot I put an 8 bar in the point group? 00:27:53.798 --> 00:27:57.510 Because it's only a fourfold rotation, and that's allowed. 00:27:57.510 --> 00:28:00.720 PROFESSOR: The reason is that there's no origin to 00:28:00.720 --> 00:28:01.650 translations. 00:28:01.650 --> 00:28:06.160 So if you had translations in an 8 bar crystal, you would 00:28:06.160 --> 00:28:11.940 have four translations that were repeated by rotation and 00:28:11.940 --> 00:28:16.920 then another translation 45 degrees and inverted down. 00:28:16.920 --> 00:28:19.760 But you can assemble those at this same point. 00:28:19.760 --> 00:28:23.710 So you would have eight translations 45 degrees apart. 00:28:23.710 --> 00:28:25.240 And that's impossible in a lattice. 00:28:30.657 --> 00:28:31.480 Need another one? 00:28:31.480 --> 00:28:32.980 AUDIENCE: I'm sorry. 00:28:32.980 --> 00:28:34.475 PROFESSOR: Do you need another one to look at? 00:28:34.475 --> 00:28:35.327 AUDIENCE: No, I'm good. 00:28:35.327 --> 00:28:36.577 PROFESSOR: OK. 00:28:39.620 --> 00:28:41.770 I can't have you just sitting there doing nothing. 00:28:46.153 --> 00:28:50.049 AUDIENCE: So is this half of this [INAUDIBLE] material? 00:28:50.049 --> 00:28:52.760 PROFESSOR: Yeah, that's twinned by a 180 degree 00:28:52.760 --> 00:28:54.872 rotation on the 1, 1, 1 plane. 00:28:54.872 --> 00:28:57.332 AUDIENCE: Oh, yeah, the 1, 1, 1 [INAUDIBLE]. 00:29:00.776 --> 00:29:02.050 PROFESSOR: Yeah. 00:29:02.050 --> 00:29:03.960 That is actually half of an octahedron. 00:29:12.590 --> 00:29:14.660 This is a-- whoops, no, it's not. 00:29:14.660 --> 00:29:15.910 Yes, it is. 00:29:25.716 --> 00:29:28.520 No, I think that's just the threefold-- 00:29:28.520 --> 00:29:30.090 3, 3, 3. 00:29:33.560 --> 00:29:35.950 3 bar, 2 over m. 00:29:35.950 --> 00:29:39.730 And it's been rotated by 180 degrees, which is not a 00:29:39.730 --> 00:29:41.450 symmetry transformation. 00:29:41.450 --> 00:29:45.535 So the 3 bar is rotated 180 degrees. 00:29:45.535 --> 00:29:47.500 AUDIENCE: That's a 3 bar, 2 over m? 00:29:47.500 --> 00:29:47.920 PROFESSOR: I think so. 00:29:47.920 --> 00:29:52.460 I think it's a 3 bar here, a face here, and a face down 00:29:52.460 --> 00:29:56.100 below, inverted, 60 degrees away. 00:29:56.100 --> 00:29:58.970 And they're mirror planes. 00:29:58.970 --> 00:30:04.785 And there two full axes perpendicular to those mirror 00:30:04.785 --> 00:30:08.110 planes, I do believe. 00:30:08.110 --> 00:30:10.490 AUDIENCE: I don't see the twofold axes. 00:30:10.490 --> 00:30:12.830 PROFESSOR: You're only seeing half the crystal, and I think 00:30:12.830 --> 00:30:14.080 that's the reason why. 00:30:21.170 --> 00:30:23.050 This is the mirror plane here. 00:30:23.050 --> 00:30:26.585 And this edge is parallel to that mirror plane. 00:30:26.585 --> 00:30:29.060 And there's a twofold axis that comes out of the middle 00:30:29.060 --> 00:30:31.230 of that edge. 00:30:31.230 --> 00:30:32.350 You can't see it. 00:30:32.350 --> 00:30:33.940 That edge. 00:30:33.940 --> 00:30:37.360 And the mirror plane is exactly parallel to that. 00:30:37.360 --> 00:30:39.868 It's tough to see because you're only seeing half of it. 00:30:39.868 --> 00:30:40.836 AUDIENCE: [INAUDIBLE]. 00:30:40.836 --> 00:30:41.320 PROFESSOR: Yeah. 00:30:41.320 --> 00:30:43.760 AUDIENCE: This [INAUDIBLE]? 00:30:43.760 --> 00:30:44.300 PROFESSOR: Yeah. 00:30:44.300 --> 00:30:46.840 This is a fourfold axis 00:30:46.840 --> 00:30:49.015 perpendicular to a mirror plane. 00:30:49.015 --> 00:30:52.420 There's a threefold axis coming out here and a twofold 00:30:52.420 --> 00:30:54.110 axis coming out here. 00:30:54.110 --> 00:30:56.740 And there is a mirror plane perpendicular to the fourfold 00:30:56.740 --> 00:31:01.180 axis and perpendicular to the twofold axis also under the 00:31:01.180 --> 00:31:01.490 mirror plane. 00:31:01.490 --> 00:31:05.876 So this is 4 over m, 3 bar, 2 over m. 00:31:05.876 --> 00:31:07.180 AUDIENCE: [INAUDIBLE]. 00:31:07.180 --> 00:31:08.780 PROFESSOR: That's the rotational symmetry. 00:31:08.780 --> 00:31:10.335 But you get all sorts of mirror planes 00:31:10.335 --> 00:31:12.120 coming through here. 00:31:12.120 --> 00:31:15.870 So really, it's this one. 00:31:15.870 --> 00:31:18.490 Mirror plane this way, mirror plane this way, 00:31:18.490 --> 00:31:19.996 mirror plane this way. 00:31:23.650 --> 00:31:30.200 So if we set it up relative to this, there's the fourfold 00:31:30.200 --> 00:31:34.700 coming out here, here's the fourfold coming out here. 00:31:34.700 --> 00:31:40.650 And there are mirror planes this way and also this way, 45 00:31:40.650 --> 00:31:41.900 degrees away. 00:31:50.915 --> 00:31:52.165 Nothing to do? 00:31:57.125 --> 00:31:58.930 Can I steal one that you're not working with? 00:31:58.930 --> 00:31:59.480 There's a-- 00:31:59.480 --> 00:32:00.730 AUDIENCE: This one is [INAUDIBLE]. 00:32:13.420 --> 00:32:18.710 PROFESSOR: That's 2, 3 with no mirror planes. 00:32:18.710 --> 00:32:22.790 Oh, no, no, that's not true. 00:32:22.790 --> 00:32:29.224 No, that is 4 bar, 2m, believe it or not. 00:32:36.610 --> 00:32:40.483 Look at the three different directions. 00:32:43.700 --> 00:32:45.620 These two corners are the same. 00:32:45.620 --> 00:32:49.790 There's a little, tiny line segment there. 00:32:49.790 --> 00:32:52.192 AUDIENCE: Yeah, so there is a mirror plane right here? 00:32:52.192 --> 00:32:54.240 PROFESSOR: Yeah, but there's no mirror plane going through 00:32:54.240 --> 00:32:56.010 the other edges. 00:32:56.010 --> 00:33:00.990 So there's a mirror plane here, mirror plane here. 00:33:00.990 --> 00:33:05.390 And then there is twofold axes coming out of this little 00:33:05.390 --> 00:33:06.970 straight line segment here. 00:33:06.970 --> 00:33:07.466 AUDIENCE: Really? 00:33:07.466 --> 00:33:07.962 PROFESSOR: Yeah. 00:33:07.962 --> 00:33:10.120 And that's different from this. 00:33:10.120 --> 00:33:13.544 So these two edges are the same. 00:33:13.544 --> 00:33:17.265 AUDIENCE: Where is the principle axis? 00:33:17.265 --> 00:33:19.550 PROFESSOR: The principle axis would be this one. 00:33:19.550 --> 00:33:20.240 AUDIENCE: This one? 00:33:20.240 --> 00:33:21.606 PROFESSOR: Yeah. 00:33:21.606 --> 00:33:27.820 AUDIENCE: So basically, I have this mirror plane right here. 00:33:27.820 --> 00:33:35.220 Then twofold axis in the middle of the circle. 00:33:35.220 --> 00:33:44.625 So now here, where is it? 00:33:44.625 --> 00:33:46.305 PROFESSOR: Well, it's hexagonal. 00:33:46.305 --> 00:33:47.890 So you want to back up a little bit. 00:33:53.236 --> 00:33:55.180 Right there. 00:33:55.180 --> 00:33:57.124 AUDIENCE: This one? 00:33:57.124 --> 00:33:58.374 PROFESSOR: No, sorry. 00:34:02.956 --> 00:34:04.920 This one. 00:34:04.920 --> 00:34:07.590 Threefold axis, twofold. 00:34:07.590 --> 00:34:09.538 That's perpendicular to a mirror plane. 00:34:09.538 --> 00:34:11.410 AUDIENCE: So here's my mirror plane. 00:34:14.218 --> 00:34:16.730 PROFESSOR: Coming right out of this little edge here. 00:34:16.730 --> 00:34:18.554 AUDIENCE: Let's say like this. 00:34:18.554 --> 00:34:19.466 No mirror plane? 00:34:19.466 --> 00:34:21.750 PROFESSOR: No. 00:34:21.750 --> 00:34:23.630 AUDIENCE: Which one is this one? 00:34:23.630 --> 00:34:25.679 It's like this one. 00:34:25.679 --> 00:34:26.900 PROFESSOR: Let's put it right here. 00:34:26.900 --> 00:34:29.726 Here is-- 00:34:29.726 --> 00:34:30.699 AUDIENCE: Like this? 00:34:30.699 --> 00:34:34.219 PROFESSOR: No, you've got to get it up like this. 00:34:34.219 --> 00:34:35.290 OK. 00:34:35.290 --> 00:34:36.830 There is a mirror plane. 00:34:36.830 --> 00:34:41.400 AUDIENCE: No, this is not. 00:34:41.400 --> 00:34:43.211 No, the mirror plane is right here. 00:34:43.211 --> 00:34:45.920 PROFESSOR: Yeah, OK. 00:34:45.920 --> 00:34:48.584 It goes up like-- 00:34:48.584 --> 00:34:50.500 oh. 00:34:50.500 --> 00:34:53.969 Let me get it set up here. 00:34:53.969 --> 00:34:54.300 OK. 00:34:54.300 --> 00:34:55.830 There is the mirror plane. 00:34:55.830 --> 00:34:59.620 This face is different from the others. 00:34:59.620 --> 00:35:03.452 AUDIENCE: And the twofold axes are on the edges. 00:35:03.452 --> 00:35:04.890 PROFESSOR: Yeah. 00:35:04.890 --> 00:35:07.437 AUDIENCE: So basically, there are three 00:35:07.437 --> 00:35:09.345 for one mirror plane. 00:35:12.210 --> 00:35:14.420 PROFESSOR: Here are the mirror planes coming through this 00:35:14.420 --> 00:35:16.760 way, this way, this way. 00:35:16.760 --> 00:35:20.080 They're 60 degrees apart. 00:35:20.080 --> 00:35:22.520 AUDIENCE: You say there are three mirror planes? 00:35:22.520 --> 00:35:25.320 PROFESSOR: Yeah, this one, this one, and this one. 00:35:25.320 --> 00:35:26.470 AUDIENCE: I don't agree. 00:35:26.470 --> 00:35:28.696 This one is no mirror plane. 00:35:28.696 --> 00:35:30.055 PROFESSOR: You're right. 00:35:30.055 --> 00:35:31.751 You're right. 00:35:31.751 --> 00:35:35.030 AUDIENCE: There is only one when you look from above. 00:35:35.030 --> 00:35:38.445 But there are two for just [INAUDIBLE] 00:35:38.445 --> 00:35:40.870 45 degrees-- or, not 45 degrees. 00:35:40.870 --> 00:35:42.120 That's a triangle. 00:35:52.040 --> 00:35:53.410 PROFESSOR: It's a terrible one. 00:36:17.218 --> 00:36:19.450 AUDIENCE: Because if we decide that the top is one, the 00:36:19.450 --> 00:36:21.020 principle-- 00:36:21.020 --> 00:36:22.420 PROFESSOR: There's a twofold axis. 00:36:22.420 --> 00:36:26.220 That's a mirror plane, and that's a mirror plane. 00:36:26.220 --> 00:36:29.140 And it looks like it is 2nn. 00:36:34.540 --> 00:36:36.761 I think it's 2nn. 00:36:36.761 --> 00:36:39.647 AUDIENCE: OK. 00:36:39.647 --> 00:36:40.609 There's nothing further? 00:36:40.609 --> 00:36:41.859 PROFESSOR: Nothing further. 00:36:45.900 --> 00:36:47.090 Got that one? 00:36:47.090 --> 00:36:49.440 AUDIENCE: 4 bar 2m? 00:36:49.440 --> 00:36:51.320 I had to look to find a twofold axis. 00:36:51.320 --> 00:36:52.270 That was the tricky part. 00:36:52.270 --> 00:36:54.388 PROFESSOR: No, no threefold axis. 00:36:54.388 --> 00:36:55.296 AUDIENCE: Well, twofold. 00:36:55.296 --> 00:36:57.112 PROFESSOR: Yeah. 00:36:57.112 --> 00:36:59.904 AUDIENCE: Finding those was [INAUDIBLE]. 00:36:59.904 --> 00:37:00.840 PROFESSOR: That's it. 00:37:00.840 --> 00:37:02.480 AUDIENCE: I can see the mirror planes. 00:37:02.480 --> 00:37:03.790 PROFESSOR: That's the twofold axis. 00:37:03.790 --> 00:37:06.730 AUDIENCE: Yeah. 00:37:06.730 --> 00:37:09.710 PROFESSOR: And this two on top and two underneath skewed by 00:37:09.710 --> 00:37:11.730 90 degrees, that's a 4 bar. 00:37:11.730 --> 00:37:13.090 So that's 4 bar 2m. 00:37:19.382 --> 00:37:22.300 You have nothing to do? 00:37:22.300 --> 00:37:23.680 Oh, you're talking with them, OK. 00:37:34.540 --> 00:37:37.650 AUDIENCE: What is this one actually? 00:37:37.650 --> 00:37:41.030 There is 4, 4, 2, 4, 3, 4, 3. 00:37:41.030 --> 00:37:43.465 PROFESSOR: Right, so it's cubic. 00:37:43.465 --> 00:37:49.540 And mirror planes are going down through the twofold axis. 00:37:49.540 --> 00:37:52.190 So that's enough to tell you that it is this. 00:37:55.320 --> 00:37:57.810 So let's set it up here. 00:37:57.810 --> 00:38:00.860 Fourfold axes are coming out of the points. 00:38:00.860 --> 00:38:03.210 So you've got mirror planes this way, 00:38:03.210 --> 00:38:05.260 mirror point 45 degrees. 00:38:05.260 --> 00:38:07.240 Here is this twofold axis. 00:38:07.240 --> 00:38:09.710 Here is this twofold axis. 00:38:09.710 --> 00:38:12.360 And here are the twofold axes that are in the same plane. 00:38:12.360 --> 00:38:13.810 And here's the other fourfold axis. 00:38:13.810 --> 00:38:15.060 AUDIENCE: I see. 00:38:24.560 --> 00:38:26.590 That one's not fair. 00:38:26.590 --> 00:38:34.180 This is actually two crystals that are grown together by an 00:38:34.180 --> 00:38:36.780 operation that is not a symmetry 00:38:36.780 --> 00:38:38.320 element of the crystal. 00:38:38.320 --> 00:38:40.430 And actually these two crystals have 00:38:40.430 --> 00:38:44.300 symmetry 2 over m. 00:38:44.300 --> 00:38:47.460 And they are rotated relative to one another by 00:38:47.460 --> 00:38:49.780 a 180 degree rotation. 00:38:49.780 --> 00:38:51.360 It's not a symmetry element. 00:38:51.360 --> 00:38:54.440 So this is something that's called a twinned crystal. 00:38:54.440 --> 00:38:54.740 AUDIENCE: Twinned? 00:38:54.740 --> 00:38:56.960 PROFESSOR: Twinned crystal. 00:38:56.960 --> 00:38:58.170 So it's really two of them. 00:38:58.170 --> 00:39:02.580 And if I could twist this one around so that the crystal 00:39:02.580 --> 00:39:04.500 continued on in that direction, it would be 00:39:04.500 --> 00:39:07.890 something that had a lozenge-like shape. 00:39:07.890 --> 00:39:09.340 So I should put that one away. 00:39:09.340 --> 00:39:11.170 That's only confusing people. 00:39:11.170 --> 00:39:12.250 AUDIENCE: Professor Wuensch? 00:39:12.250 --> 00:39:12.620 PROFESSOR: Yes, sir. 00:39:12.620 --> 00:39:13.870 AUDIENCE: Is that a 6 over mmm? 00:39:17.350 --> 00:39:21.090 PROFESSOR: Yes, that's 6 over m, 2 over m, 2 over m. 00:39:21.090 --> 00:39:22.410 Six folds. 00:39:22.410 --> 00:39:24.730 Two kinds of twofold, one out of the edge, 00:39:24.730 --> 00:39:26.880 one out of the corner. 00:39:26.880 --> 00:39:27.810 And a mirror plane 00:39:27.810 --> 00:39:30.810 perpendicular to each of those. 00:39:30.810 --> 00:39:32.650 6 over m, 2 over m, 2 over m. 00:39:32.650 --> 00:39:36.386 AUDIENCE: So that's the same thing [INAUDIBLE]. 00:39:40.494 --> 00:39:42.979 So the 6 over m comes from this way. 00:39:42.979 --> 00:39:45.464 And then the 2 over m is there. 00:39:45.464 --> 00:39:47.452 AUDIENCE: Yeah, so it's [INAUDIBLE]. 00:39:47.452 --> 00:39:48.702 AUDIENCE: Yeah, exactly. 00:39:52.440 --> 00:39:53.150 PROFESSOR: Let me put that one away. 00:39:53.150 --> 00:39:54.700 It's just confusing people. 00:39:54.700 --> 00:39:55.460 AUDIENCE: Oh, that one was easy. 00:39:55.460 --> 00:39:56.800 PROFESSOR: That's a twinned crystal. 00:39:56.800 --> 00:39:57.240 It was easy? 00:39:57.240 --> 00:39:57.920 You found it easy? 00:39:57.920 --> 00:39:59.610 AUDIENCE: Yeah, so 3 over m, right? 00:40:03.270 --> 00:40:04.520 PROFESSOR: Yeah, I guess. 00:40:10.120 --> 00:40:12.850 This looks as though it might be the corner of an octahedron 00:40:12.850 --> 00:40:13.920 because you don't see much of it. 00:40:13.920 --> 00:40:17.490 But these two faces and these two faces are not related by a 00:40:17.490 --> 00:40:18.740 fourfold rotation. 00:40:20.600 --> 00:40:22.370 AUDIENCE: Well, either way, there wouldn't be anywhere to 00:40:22.370 --> 00:40:23.330 put a fourfold rotation. 00:40:23.330 --> 00:40:24.760 PROFESSOR: Yeah, right. 00:40:24.760 --> 00:40:27.590 So if you look at the whole thing, this is actually two 00:40:27.590 --> 00:40:29.690 intergrown crystals when you see this sort 00:40:29.690 --> 00:40:31.110 of reentrant angle. 00:40:31.110 --> 00:40:36.570 This is two crystals grown together by an operation which 00:40:36.570 --> 00:40:39.252 is not a symmetry operation. 00:40:39.252 --> 00:40:40.797 So this is something that's called a 00:40:40.797 --> 00:40:42.420 twin, a twinned crystal. 00:40:42.420 --> 00:40:43.830 AUDIENCE: But the symmetry is 00:40:43.830 --> 00:40:45.650 nevertheless 3 over m, correct? 00:40:45.650 --> 00:40:46.330 PROFESSOR: Of the whole thing? 00:40:46.330 --> 00:40:47.550 Yeah, yeah, of the whole thing. 00:40:47.550 --> 00:40:50.100 AUDIENCE: And this is just 3. 00:40:50.100 --> 00:40:51.750 PROFESSOR: No, this is 3mm. 00:40:56.080 --> 00:40:59.075 There's a mirror plane down there and a 00:40:59.075 --> 00:41:00.210 mirror plane down here. 00:41:00.210 --> 00:41:01.020 AUDIENCE: You're right. 00:41:01.020 --> 00:41:03.980 PROFESSOR: No twofold axis because this end is different 00:41:03.980 --> 00:41:06.100 from this end. 00:41:06.100 --> 00:41:07.580 AUDIENCE: So this is a regular solid, right? 00:41:07.580 --> 00:41:09.050 What is it called? 00:41:09.050 --> 00:41:12.188 PROFESSOR: This is a rhombic dodecahedron. 00:41:12.188 --> 00:41:14.164 AUDIENCE: Oh, the one you were talking about the other day. 00:41:14.164 --> 00:41:14.660 PROFESSOR: Yeah. 00:41:14.660 --> 00:41:19.640 This is 4, 3, 2. 00:41:19.640 --> 00:41:21.890 And then mirror planes too all over the place. 00:41:21.890 --> 00:41:23.630 AUDIENCE: So all of these solids do not have to be 00:41:23.630 --> 00:41:25.840 members of this list, right? 00:41:25.840 --> 00:41:28.460 Because these are finite objects. 00:41:28.460 --> 00:41:30.610 They can be [? twelvefold ?] and-- 00:41:30.610 --> 00:41:31.290 PROFESSOR: They could be. 00:41:31.290 --> 00:41:34.110 These actually, in fact, are models of 00:41:34.110 --> 00:41:35.820 real crystalline materials. 00:41:35.820 --> 00:41:39.490 And they're made with the faces and the relative sizes 00:41:39.490 --> 00:41:41.835 that these minerals actually have. 00:41:41.835 --> 00:41:43.760 AUDIENCE: OK. 00:41:43.760 --> 00:41:46.020 PROFESSOR: So somewhere floating around is something 00:41:46.020 --> 00:41:48.830 that's very characteristically quartz. 00:41:48.830 --> 00:41:52.095 And there's one that's a flat one with a reentrant angle in 00:41:52.095 --> 00:41:53.345 it, that's gypsum. 00:41:59.170 --> 00:42:01.080 AUDIENCE: [INAUDIBLE]. 00:42:01.080 --> 00:42:03.820 PROFESSOR: Yeah, that's fourfold. 00:42:03.820 --> 00:42:05.560 Twofold here. 00:42:05.560 --> 00:42:07.900 This goes into this. 00:42:07.900 --> 00:42:10.740 No mirror planes because these things are inclined. 00:42:10.740 --> 00:42:15.942 So it's 4, 2, 2. 00:42:15.942 --> 00:42:17.192 Good. 00:42:19.770 --> 00:42:21.430 It's easy when you know what to look for. 00:42:21.430 --> 00:42:24.010 When you say, they are only a few possibilities. 00:42:24.010 --> 00:42:27.630 And if I see a fourfold axis in it, that narrows it down to 00:42:27.630 --> 00:42:28.880 two or three. 00:42:43.385 --> 00:42:43.870 AUDIENCE: Like that. 00:42:43.870 --> 00:42:48.525 That's an inversion center running through here. 00:42:48.525 --> 00:42:51.614 And there's no rotation there [INAUDIBLE]. 00:42:51.614 --> 00:42:53.606 AUDIENCE: [INAUDIBLE] here? 00:42:53.606 --> 00:42:55.598 Here through this way? 00:42:58.586 --> 00:43:02.072 AUDIENCE: No, because this doesn't map [INAUDIBLE]. 00:43:05.060 --> 00:43:09.044 Does that map that by a mirror? 00:43:09.044 --> 00:43:11.036 The mirror [INAUDIBLE]? 00:43:11.036 --> 00:43:13.526 AUDIENCE: It cuts across this way. 00:43:13.526 --> 00:43:16.514 AUDIENCE: Yeah, I could buy that. 00:43:19.530 --> 00:43:23.070 PROFESSOR: You've reached a consensus? 00:43:23.070 --> 00:43:24.690 AUDIENCE: We've got a mirror, and we've got an inversion. 00:43:24.690 --> 00:43:27.240 But we're not sure what else we've got. 00:43:27.240 --> 00:43:30.471 AUDIENCE: While I like the inversion, I'd 00:43:30.471 --> 00:43:31.721 also like the mirror. 00:43:39.990 --> 00:43:41.240 I don't see the mirror. 00:43:49.492 --> 00:43:51.977 All I see is a really an inversion axis there. 00:43:51.977 --> 00:43:55.960 I think I'm missing that mirror that you're saying. 00:43:55.960 --> 00:43:58.610 PROFESSOR: There's a mirror that goes down this way. 00:43:58.610 --> 00:44:00.800 And there's a twofold axis here. 00:44:03.395 --> 00:44:04.790 AUDIENCE: OK, that was the one we missed. 00:44:04.790 --> 00:44:06.570 PROFESSOR: So that's 2 over m. 00:44:06.570 --> 00:44:09.340 AUDIENCE: Over m. 00:44:09.340 --> 00:44:10.860 And the inversion just falls out. 00:44:10.860 --> 00:44:13.050 PROFESSOR: Yeah, that's in there too. 00:44:13.050 --> 00:44:15.980 Inversions are nice. 00:44:15.980 --> 00:44:17.340 You can feel inversions. 00:44:17.340 --> 00:44:20.160 You put your finger on one face and your finger on the 00:44:20.160 --> 00:44:21.380 other face. 00:44:21.380 --> 00:44:22.630 [INTERPOSING VOICES] 00:44:39.092 --> 00:44:39.584 PROFESSOR: Get it? 00:44:39.584 --> 00:44:40.950 AUDIENCE: So this is this one, right? 00:44:40.950 --> 00:44:43.890 PROFESSOR: Yes, very good. 00:44:43.890 --> 00:44:47.865 And that is a silicate mineral called garnet. 00:44:47.865 --> 00:44:48.820 AUDIENCE: What's that? 00:44:48.820 --> 00:44:50.060 PROFESSOR: Garnet. 00:44:50.060 --> 00:44:54.030 It's actually a silicate mineral, which is used as a 00:44:54.030 --> 00:44:55.280 gemstone sometimes. 00:45:08.526 --> 00:45:10.490 AUDIENCE: So this should be a 4, 3, 2 face. 00:45:13.436 --> 00:45:14.418 PROFESSOR: Yep. 00:45:14.418 --> 00:45:15.891 AUDIENCE: But I don't know if it's the end of the 00:45:15.891 --> 00:45:17.141 [INAUDIBLE]. 00:45:21.783 --> 00:45:24.238 Could be m3m. 00:45:24.238 --> 00:45:25.488 AUDIENCE: Where are the-- 00:45:28.170 --> 00:45:29.610 it doesn't an inversion center. 00:45:29.610 --> 00:45:30.296 PROFESSOR: Hmm? 00:45:30.296 --> 00:45:32.470 AUDIENCE: It doesn't have an inversion center here. 00:45:32.470 --> 00:45:33.140 AUDIENCE: Oh yeah, it does. 00:45:33.140 --> 00:45:33.790 PROFESSOR: Oh yes, it does. 00:45:33.790 --> 00:45:36.060 Yes, it does. 00:45:36.060 --> 00:45:37.430 Actually, the thing to look for are the 00:45:37.430 --> 00:45:38.290 high symmetry things. 00:45:38.290 --> 00:45:39.350 Your eye spots them. 00:45:39.350 --> 00:45:41.720 That's got a fourfold axis in it. 00:45:41.720 --> 00:45:43.440 OK 00:45:43.440 --> 00:45:46.140 So are there other fourfold axes? 00:45:46.140 --> 00:45:47.720 Yes, yes. 00:45:47.720 --> 00:45:50.930 So that has to be based on 4, 3, 2. 00:45:50.930 --> 00:45:52.970 Because you have three orthogonal fourfold axes. 00:45:52.970 --> 00:45:54.710 Threefold is here. 00:45:54.710 --> 00:45:55.860 Twofold axis is here. 00:45:55.860 --> 00:45:57.110 AUDIENCE: OK. 00:46:06.045 --> 00:46:07.295 PROFESSOR: OK, thanks. 00:46:09.588 --> 00:46:11.049 AUDIENCE: There are-- 00:46:11.049 --> 00:46:12.742 oh, I already saw those ones. 00:46:12.742 --> 00:46:14.370 PROFESSOR: You want to look at some more? 00:46:14.370 --> 00:46:15.170 AUDIENCE: All right. 00:46:15.170 --> 00:46:16.420 PROFESSOR: Enough for one day? 00:46:21.696 --> 00:46:23.687 AUDIENCE: Professor Wuensch, I've been staring at this one 00:46:23.687 --> 00:46:24.735 for such a long time. 00:46:24.735 --> 00:46:28.400 And I couldn't match it with anything on here. 00:46:28.400 --> 00:46:34.090 PROFESSOR: This looks like a tetrahedron except the face is 00:46:34.090 --> 00:46:35.340 puckered up into this little thing. 00:46:35.340 --> 00:46:37.810 So the way I would start with saying, OK, this is 00:46:37.810 --> 00:46:41.880 tetrahedral in which case if I make these things flat, that's 00:46:41.880 --> 00:46:43.130 a perfect tetrahedron. 00:46:43.130 --> 00:46:44.500 Four sides. 00:46:44.500 --> 00:46:46.590 But what's happened is that this face is 00:46:46.590 --> 00:46:47.840 not quite 1, 1, 1. 00:46:50.800 --> 00:46:55.740 The 4 bar axes come out these three directions. 00:46:55.740 --> 00:47:00.220 And they have got twofold symmetry, mirror planes 00:47:00.220 --> 00:47:02.550 running through the twofold axes. 00:47:02.550 --> 00:47:05.940 This is the 4 bar axis with mirror planes running this 00:47:05.940 --> 00:47:07.490 way, this way. 00:47:07.490 --> 00:47:12.603 So this is actually 4 bar. 00:47:20.300 --> 00:47:21.840 This is still further. 00:47:21.840 --> 00:47:23.090 It's a cubic crystal. 00:47:26.850 --> 00:47:31.360 This is based on the tetrahedral symmetry. 00:47:31.360 --> 00:47:35.115 Here is the 4 bar axis. 00:47:39.530 --> 00:47:43.460 Mirror plane this way, mirror plane this way. 00:47:43.460 --> 00:47:47.260 Another 4 bar coming out of these two edges. 00:47:47.260 --> 00:47:50.020 And then these are the diagonal mirror planes. 00:47:50.020 --> 00:47:53.635 So this is it. 00:47:53.635 --> 00:47:56.373 AUDIENCE: I think I was looking for a threefold axis 00:47:56.373 --> 00:47:57.100 in the center. 00:47:57.100 --> 00:48:02.720 PROFESSOR: No, the 4 bar comes out of the edge of the 00:48:02.720 --> 00:48:03.970 tetrahedron. 00:48:05.600 --> 00:48:09.690 And this is the other 4 bar coming out here. 00:48:09.690 --> 00:48:13.270 Threefold like this and mirror planes going through the 00:48:13.270 --> 00:48:15.312 threefold axis like that. 00:48:15.312 --> 00:48:18.230 AUDIENCE: I see it. 00:48:18.230 --> 00:48:19.860 PROFESSOR: It's hard when you look at this thing. 00:48:19.860 --> 00:48:21.060 I never saw this thing before. 00:48:21.060 --> 00:48:22.020 What's here? 00:48:22.020 --> 00:48:25.410 But if you know the results have to be 1 of these 32 00:48:25.410 --> 00:48:27.580 possibilities, and this is obviously cubic, and it's 00:48:27.580 --> 00:48:30.580 based on the tetrahedron, that means there are only three 00:48:30.580 --> 00:48:32.350 possibilities. 00:48:32.350 --> 00:48:33.820 No mirror planes. 00:48:33.820 --> 00:48:37.110 If you find one mirror plane, you ask yourself, does the 00:48:37.110 --> 00:48:39.590 mirror plane go through the threefold axis, or does it 00:48:39.590 --> 00:48:41.110 miss the threefold axis? 00:48:41.110 --> 00:48:43.930 It goes through the threefold axis, so it's got to be this. 00:48:43.930 --> 00:48:47.500 And then you know just what to look for. 00:48:47.500 --> 00:48:51.650 AUDIENCE: Is it just [INAUDIBLE] m? 00:48:51.650 --> 00:48:53.120 PROFESSOR: This is a dirty one. 00:48:53.120 --> 00:48:54.130 This is a dirty one. 00:48:54.130 --> 00:48:58.140 If you look at this very carefully, this face is the 00:48:58.140 --> 00:49:01.265 same as this face, but it's not the same as that one. 00:49:01.265 --> 00:49:02.850 It's a little bigger. 00:49:02.850 --> 00:49:07.760 So if you say, what's in here, these two things do not come 00:49:07.760 --> 00:49:11.400 together at a common vertex like these two. 00:49:11.400 --> 00:49:15.250 There's another little line segment between these faces. 00:49:15.250 --> 00:49:18.380 So there are no threefold axes coming out of this. 00:49:18.380 --> 00:49:20.230 Looks like it might be tetrahedral. 00:49:20.230 --> 00:49:23.070 Clearly, a mirror plane going this way. 00:49:23.070 --> 00:49:30.180 And then there is also a twofold axis coming out here. 00:49:30.180 --> 00:49:37.430 So there's a 2 over m and another 2 over m. 00:49:47.440 --> 00:49:50.890 I did this once before with somebody, and this is the 00:49:50.890 --> 00:49:52.678 hardest one in the whole set. 00:50:02.305 --> 00:50:03.780 So a mirror plane this way. 00:50:06.710 --> 00:50:08.570 There's got to be another mirror plane because there's a 00:50:08.570 --> 00:50:09.820 twofold axis. 00:50:17.440 --> 00:50:19.162 4 bar. 00:50:19.162 --> 00:50:24.940 Up, down, up, down. 00:50:24.940 --> 00:50:31.878 So this is 4 bar, 2 over m. 00:50:34.560 --> 00:50:35.810 4 bar, 2 over m.