1 00:00:12,980 --> 00:00:15,670 PROFESSOR: Any questions before I obliterate all this 2 00:00:15,670 --> 00:00:16,920 lovely geometry? 3 00:00:19,100 --> 00:00:20,120 No. 4 00:00:20,120 --> 00:00:20,410 OK. 5 00:00:20,410 --> 00:00:22,910 We handled them during intermission I guess. 6 00:00:22,910 --> 00:00:27,370 Let me do a couple more plane groups just very quickly to 7 00:00:27,370 --> 00:00:30,330 show you how they come out without going through all of 8 00:00:30,330 --> 00:00:33,900 the steps, because I think we've seen now what one has to 9 00:00:33,900 --> 00:00:35,520 do to derive these. 10 00:00:35,520 --> 00:00:37,050 There are two that are left. 11 00:00:37,050 --> 00:00:42,140 One is a threefold axis, and if that's all the symmetry 12 00:00:42,140 --> 00:00:46,210 we've put into the lattice, we're combining a threefold 13 00:00:46,210 --> 00:00:50,030 rotation axis with a primitive lattice, and we know that this 14 00:00:50,030 --> 00:00:51,310 has to be hexagonal net. 15 00:00:54,110 --> 00:00:57,490 Because from our depth of experience, we know that is 16 00:00:57,490 --> 00:01:01,020 the shape of a lattice that is demanded by a threefold axis. 17 00:01:06,850 --> 00:01:11,750 It has two translations, T1, that are identical in length. 18 00:01:11,750 --> 00:01:15,940 And this angle between them is exactly 120 degrees. 19 00:01:15,940 --> 00:01:19,480 That's what we saw a threefold access require. 20 00:01:19,480 --> 00:01:23,100 So now what we are doing is putting in a threefold axis at 21 00:01:23,100 --> 00:01:25,950 one lattice point, and this means we are adding the 22 00:01:25,950 --> 00:01:30,330 operations A 2 pi over 3. 23 00:01:30,330 --> 00:01:33,170 A minus 2 pi over 3. 24 00:01:33,170 --> 00:01:37,600 I'll choose to define a 120 degree rotation that way, and 25 00:01:37,600 --> 00:01:41,215 the operation A 2 pi, which is the identity operations. 26 00:01:41,215 --> 00:01:42,660 So I'm adding three operations. 27 00:01:47,260 --> 00:01:52,990 Putting the threefold axis in at all these locations. 28 00:01:52,990 --> 00:01:57,000 They will all be translationally equivalent. 29 00:01:57,000 --> 00:02:00,820 The pattern of this particular plane group is going to be the 30 00:02:00,820 --> 00:02:03,290 pattern of a threefold axis. 31 00:02:03,290 --> 00:02:07,560 And so there will be one motif here, 120 degrees away. 32 00:02:07,560 --> 00:02:11,500 There will be a-- 33 00:02:11,500 --> 00:02:15,360 sorry to wipe out those operations-- 34 00:02:15,360 --> 00:02:16,570 120 degrees. 35 00:02:16,570 --> 00:02:21,430 There'll be another object located here. 36 00:02:24,440 --> 00:02:26,870 And 120 degrees again, there will be another 37 00:02:26,870 --> 00:02:28,210 object located here. 38 00:02:28,210 --> 00:02:30,770 So these guys will fall at the corner of 39 00:02:30,770 --> 00:02:34,070 an equilateral triangle. 40 00:02:34,070 --> 00:02:38,320 And the same will be at the other corners of the net, and 41 00:02:38,320 --> 00:02:41,600 that, as I've said before, is the pattern 42 00:02:41,600 --> 00:02:42,760 of that plane group. 43 00:02:42,760 --> 00:02:47,440 And that's a plane group that we would call P3. 44 00:02:47,440 --> 00:02:50,880 Primitive lattice that has to be hexagonal, and what we've 45 00:02:50,880 --> 00:02:54,560 added to it is a threefold axis. 46 00:02:54,560 --> 00:02:57,870 OK, let me down here just indicate the combinations that 47 00:02:57,870 --> 00:02:58,480 we would do. 48 00:02:58,480 --> 00:03:00,900 And I could do it up on top, but I can work with this 49 00:03:00,900 --> 00:03:04,210 translation as well, which is easier to reach at this late 50 00:03:04,210 --> 00:03:05,650 hour in the afternoon. 51 00:03:05,650 --> 00:03:11,790 Let me combine A 2 pi over 3 with this translation here. 52 00:03:11,790 --> 00:03:17,060 And that says we have to get an operation B 2 pi over 3. 53 00:03:17,060 --> 00:03:21,880 That is located at the original translation T times 54 00:03:21,880 --> 00:03:28,820 the cotangent of 1/2 of 2 pi over 3. 55 00:03:28,820 --> 00:03:33,670 And that's at 1/2 of the cotangent of 60 degrees. 56 00:03:33,670 --> 00:03:38,640 And this turns out not to be any nice neat number like 0 or 57 00:03:38,640 --> 00:03:44,000 1/2, but if you evaluate what the distance up along the 58 00:03:44,000 --> 00:03:47,880 perpendicular bisector is by this amount, where you come 59 00:03:47,880 --> 00:03:52,820 out is right in the center of this triangle. 60 00:03:52,820 --> 00:03:53,890 Trust me. 61 00:03:53,890 --> 00:03:56,000 A little bit of trigonometry will let you 62 00:03:56,000 --> 00:03:59,090 convince yourself of that. 63 00:03:59,090 --> 00:04:05,760 So if I rotate 120 degrees. 64 00:04:05,760 --> 00:04:11,620 Bring this one to this one, and then translate over to 65 00:04:11,620 --> 00:04:18,959 here, the way the first one and the second one are related 66 00:04:18,959 --> 00:04:24,020 is by a 120 degree rotation about the 67 00:04:24,020 --> 00:04:25,270 center of this triangle. 68 00:04:27,760 --> 00:04:32,330 If I combine the operation A minus 2 pi over 3, with this 69 00:04:32,330 --> 00:04:35,740 translation, I go down a distance x of minus this 70 00:04:35,740 --> 00:04:40,420 amount and that puts me in the middle of the triangle that is 71 00:04:40,420 --> 00:04:42,250 directly below. 72 00:04:42,250 --> 00:04:44,370 And I can move that back up. 73 00:04:44,370 --> 00:04:48,320 And now I have all the operations of a threefold axis 74 00:04:48,320 --> 00:04:51,660 about this location. 75 00:04:51,660 --> 00:04:55,730 I can do the same thing with the threefold axis at another 76 00:04:55,730 --> 00:04:57,160 lattice point. 77 00:04:57,160 --> 00:04:58,830 Another thing I could do is to just 78 00:04:58,830 --> 00:05:01,560 rotate this by 120 degrees. 79 00:05:01,560 --> 00:05:05,260 And by either route, you find there must be a 80 00:05:05,260 --> 00:05:06,560 threefold axis here. 81 00:05:09,370 --> 00:05:12,680 Notice that these threefold axes will once again, as 82 00:05:12,680 --> 00:05:18,490 advertised, simply take things that are at the different 83 00:05:18,490 --> 00:05:20,210 corners of the cell. 84 00:05:20,210 --> 00:05:24,590 For example, this threefold axis will tell you how this 85 00:05:24,590 --> 00:05:29,700 motif is related to this one is related to this one. 86 00:05:29,700 --> 00:05:36,130 And so it goes through the other threefold axes as well. 87 00:05:36,130 --> 00:05:39,020 This threefold axis will tell you this one is related. 88 00:05:39,020 --> 00:05:41,020 No, I don't want to draw it in. 89 00:05:41,020 --> 00:05:45,500 So this is P3, a pair of threefold axes in the centers 90 00:05:45,500 --> 00:05:49,120 of the triangle, and another one at the corner of the cell. 91 00:05:52,620 --> 00:05:56,030 And now I am going to do the remaining combination of a 92 00:05:56,030 --> 00:06:00,580 rotation axis with a lattice, so quickly, and it's going to 93 00:06:00,580 --> 00:06:03,650 take your breath away. 94 00:06:03,650 --> 00:06:06,480 And that is seemingly the most difficult and 95 00:06:06,480 --> 00:06:08,170 complex one of all. 96 00:06:13,300 --> 00:06:16,490 This would be P6. 97 00:06:16,490 --> 00:06:19,860 Sixfold axis plus a primitive lattice. 98 00:06:19,860 --> 00:06:23,010 And we know it also has to be hexagonal. 99 00:06:23,010 --> 00:06:25,950 And that will be called P6. 100 00:06:25,950 --> 00:06:28,970 What we're adding to the lattice is a sixfold axis. 101 00:06:36,500 --> 00:06:39,490 And now what I'm going to do as a shortcut is to say a 102 00:06:39,490 --> 00:06:44,340 sixfold axis also contains all the operations 103 00:06:44,340 --> 00:06:45,730 of a threefold axis. 104 00:06:45,730 --> 00:06:51,150 So I can take P3 and drop it right on top of P6, and that's 105 00:06:51,150 --> 00:06:53,980 going to give me threefold axes here. 106 00:06:53,980 --> 00:06:59,320 Sixfold axis not only contains 2 pi over 6 and 2 pi over 3, 107 00:06:59,320 --> 00:07:03,230 it also contains the operation 2 pi over 2. 108 00:07:03,230 --> 00:07:06,460 This is the operation of a twofold axis, and that says I 109 00:07:06,460 --> 00:07:09,150 have to, in addition to the two full rotation that sits 110 00:07:09,150 --> 00:07:13,730 here get twofold rotations in the middle of every one of the 111 00:07:13,730 --> 00:07:16,830 translations T2. 112 00:07:16,830 --> 00:07:20,555 So actually, this is going to be P2 superimposed on P3. 113 00:07:20,555 --> 00:07:22,750 And that's going to give me these twofold axis and this 114 00:07:22,750 --> 00:07:23,700 threefold axis. 115 00:07:23,700 --> 00:07:28,520 And the only thing I have to do is to show you what A 2 pi 116 00:07:28,520 --> 00:07:31,120 over 6 combined with one of these 117 00:07:31,120 --> 00:07:33,330 translations is going to do. 118 00:07:33,330 --> 00:07:37,350 What is A 2 pi over 6 combined with this translation? 119 00:07:37,350 --> 00:07:41,850 And it's going to be new operation B 2 pi over 6, and 120 00:07:41,850 --> 00:07:49,990 it's going to be located at 1/2 of T times the cotangent 121 00:07:49,990 --> 00:07:54,400 of 1/2 of 60 degrees, cotangent of 30 degrees. 122 00:07:54,400 --> 00:08:03,640 And the cotangent of 30 degrees is 2. 123 00:08:03,640 --> 00:08:07,800 So this is going to be at 1/2-- 124 00:08:07,800 --> 00:08:09,720 no, what do I want to say? 125 00:08:09,720 --> 00:08:11,260 This is 30 degrees. 126 00:08:11,260 --> 00:08:12,030 This is one. 127 00:08:12,030 --> 00:08:13,430 This is two. 128 00:08:13,430 --> 00:08:18,080 Cotangent of 30 degrees is this over this, and that is-- 129 00:08:21,360 --> 00:08:22,610 no that's one. 130 00:08:26,540 --> 00:08:28,020 AUDIENCE: It's 2 pi [? squared 3. ?] 131 00:08:28,020 --> 00:08:28,360 PROFESSOR: Yeah. 132 00:08:28,360 --> 00:08:29,180 OK, that's right. 133 00:08:29,180 --> 00:08:33,039 So actually, what that does is to say the sixfold axis sits 134 00:08:33,039 --> 00:08:35,390 right up here. 135 00:08:35,390 --> 00:08:40,260 So I don't get any new sixfold axis rotation of 60 degrees 136 00:08:40,260 --> 00:08:43,100 here, followed by translation is the same as 60 degrees 137 00:08:43,100 --> 00:08:44,770 about here. 138 00:08:44,770 --> 00:08:51,430 So this is P6, and there is a lot of pure rotation axes 139 00:08:51,430 --> 00:08:53,560 combined with lattices. 140 00:08:53,560 --> 00:08:59,240 We've got P1, P2, P3, P4, and P6. 141 00:09:06,310 --> 00:09:08,770 Now, what I will do eventually, when we're all 142 00:09:08,770 --> 00:09:15,990 done here, the plane groups are not derived in any book or 143 00:09:15,990 --> 00:09:18,650 set of tables that I am aware of. 144 00:09:18,650 --> 00:09:21,830 And next time, you will get some notes that do this in 145 00:09:21,830 --> 00:09:26,450 very slow motion fashion and give you all 146 00:09:26,450 --> 00:09:28,510 the individual steps. 147 00:09:28,510 --> 00:09:33,120 But the international tables does give you diagrams of the 148 00:09:33,120 --> 00:09:38,160 resulting plane groups, very nice carefully done figures 149 00:09:38,160 --> 00:09:40,830 along with the representative arrangement of motifs that 150 00:09:40,830 --> 00:09:43,160 they generate. 151 00:09:43,160 --> 00:09:44,620 But we're not done yet. 152 00:09:44,620 --> 00:09:47,800 We have not let mirror planes enter the picture. 153 00:09:53,210 --> 00:09:57,790 And so unless there is dissension or debate, I'd like 154 00:09:57,790 --> 00:10:02,350 to consider what happens when we take a mirror plane, and we 155 00:10:02,350 --> 00:10:07,450 can combine that with two different kinds of lattices, a 156 00:10:07,450 --> 00:10:12,620 primitive rectangular net or a centered rectangular net, 157 00:10:12,620 --> 00:10:21,250 which is called C. And in order to do that, we need yet 158 00:10:21,250 --> 00:10:23,340 another combination theorem. 159 00:10:29,510 --> 00:10:36,280 Here are the lattice points, and let me first derive the 160 00:10:36,280 --> 00:10:39,860 plane group that is called PM. 161 00:10:39,860 --> 00:10:43,040 And what I'll do is to put the-- 162 00:10:43,040 --> 00:10:48,050 let me use a squiggly line here, not because I'm excited 163 00:10:48,050 --> 00:10:50,740 or nervous about this, but just to distinguish it from 164 00:10:50,740 --> 00:10:52,240 the edges of the cell. 165 00:10:52,240 --> 00:10:54,760 So here is the operation sigma. 166 00:10:54,760 --> 00:10:59,080 The pattern that is going to be displayed by the plane 167 00:10:59,080 --> 00:11:02,670 group is once again just the pattern produced by a mirror 168 00:11:02,670 --> 00:11:04,460 plane hung at every lattice point. 169 00:11:07,050 --> 00:11:09,520 But now we need a theorem. 170 00:11:09,520 --> 00:11:15,320 What we have here is the operation of reflection, 171 00:11:15,320 --> 00:11:21,600 followed by a translation that is perpendicular to the locus 172 00:11:21,600 --> 00:11:23,330 of the reflection line. 173 00:11:23,330 --> 00:11:26,700 And we will ask what is that? 174 00:11:30,610 --> 00:11:33,580 Again, you get the answer by just looking at once and for 175 00:11:33,580 --> 00:11:36,050 all, and say, if here is a first one and it is 176 00:11:36,050 --> 00:11:40,020 right-handed, and I reflect it to one number to a second one, 177 00:11:40,020 --> 00:11:43,630 number two, which is left-handed, and then move 178 00:11:43,630 --> 00:11:48,390 that by translation here to get number three, which stays 179 00:11:48,390 --> 00:11:51,120 left-handed if I move by translation. 180 00:11:51,120 --> 00:11:55,160 And ask now how was one related to three. 181 00:11:55,160 --> 00:11:58,050 The chirality is changed. 182 00:11:58,050 --> 00:12:00,970 Reflection is the only thing available to us, and lo and 183 00:12:00,970 --> 00:12:05,250 behold, if I say there is a new mirror plane here, that 184 00:12:05,250 --> 00:12:07,800 tells me how this is related to this, and this one is 185 00:12:07,800 --> 00:12:08,890 related to this one. 186 00:12:08,890 --> 00:12:15,270 And this to this and this to this, so the answer to this 187 00:12:15,270 --> 00:12:18,780 question is that a reflection combined with a perpendicular 188 00:12:18,780 --> 00:12:21,850 translation is a new reflection operation sigma 189 00:12:21,850 --> 00:12:27,690 prime that is located at a distance removed from the 190 00:12:27,690 --> 00:12:33,230 first by 1/2 of that perpendicular translation. 191 00:12:33,230 --> 00:12:35,530 And again, it's a plane old mirror plane just like the 192 00:12:35,530 --> 00:12:38,950 first one, but notice that the disposition of objects 193 00:12:38,950 --> 00:12:42,230 relative to the mirror plane in the center of the cell is 194 00:12:42,230 --> 00:12:45,530 quite distinct from the disposition of objects 195 00:12:45,530 --> 00:12:48,000 relative to the first mirror plane, so this is a second 196 00:12:48,000 --> 00:12:48,960 mirror plane. 197 00:12:48,960 --> 00:12:56,220 It is an independent mirror plane from the first. 198 00:12:56,220 --> 00:13:00,710 So that is plane group PM. 199 00:13:00,710 --> 00:13:03,360 If there's symmetry in this business, you might ask is 200 00:13:03,360 --> 00:13:07,080 there a plane group AM? 201 00:13:07,080 --> 00:13:09,180 The answer is yes in three dimensions. 202 00:13:09,180 --> 00:13:14,070 There is a space group, PM, and there's a space group AM. 203 00:13:14,070 --> 00:13:16,960 So there's AM and PM, and there is symmetry, and all is 204 00:13:16,960 --> 00:13:18,210 well in the universe. 205 00:13:20,880 --> 00:13:23,380 OK, we're making great progress here. 206 00:13:23,380 --> 00:13:30,540 And we'll be fairly well along before we have to bring things 207 00:13:30,540 --> 00:13:32,230 to a close. 208 00:13:32,230 --> 00:13:34,490 Let's do the second addition that's possible 209 00:13:34,490 --> 00:13:37,440 with a mirror plane. 210 00:13:37,440 --> 00:13:42,120 And that is to take a reflection operation and add 211 00:13:42,120 --> 00:13:46,360 it to the translations that are present in a centered 212 00:13:46,360 --> 00:13:47,610 rectangular net. 213 00:13:49,720 --> 00:13:54,560 We've already done all the work for PM, so we can use 214 00:13:54,560 --> 00:13:57,410 that as a starting point. 215 00:13:57,410 --> 00:14:00,880 The pattern of this plane group is going to look like a 216 00:14:00,880 --> 00:14:04,510 pair of objects related by reflection. 217 00:14:04,510 --> 00:14:07,850 But now, we'll have an extra pair hung at the centered 218 00:14:07,850 --> 00:14:09,100 lattice point as well. 219 00:14:13,280 --> 00:14:17,230 And the first thing we can note is that this mirror plane 220 00:14:17,230 --> 00:14:19,980 is no longer independent of the first one. 221 00:14:30,100 --> 00:14:33,990 What goes on at this mirror plane is something that's 222 00:14:33,990 --> 00:14:37,490 related to what happens at the origin lattice point and 223 00:14:37,490 --> 00:14:40,630 mirror plane, and therefore, these two mirror planes are 224 00:14:40,630 --> 00:14:42,670 going to do the same thing. 225 00:14:42,670 --> 00:14:48,660 The pattern of this plane group, we've taken a mirror 226 00:14:48,660 --> 00:14:53,800 plane and added it to a centered rectangular net. 227 00:14:53,800 --> 00:14:57,540 This is called a C lattice, standing for centered. 228 00:14:57,540 --> 00:15:01,850 And correspondingly, the symbol for this 229 00:15:01,850 --> 00:15:03,100 plane group is CM. 230 00:15:06,790 --> 00:15:08,710 We know how this one is related to this one. 231 00:15:08,710 --> 00:15:10,440 This one related to this one. 232 00:15:10,440 --> 00:15:14,700 And now, we've got something of a problem. 233 00:15:14,700 --> 00:15:18,670 All of these motifs are equivalent. 234 00:15:18,670 --> 00:15:22,140 How is this one related to this one? 235 00:15:25,520 --> 00:15:29,890 Or in more general terms, what we're asking is suppose I have 236 00:15:29,890 --> 00:15:35,900 a reflection operation sigma, and I add the translation not 237 00:15:35,900 --> 00:15:44,290 at right angles to it, as I did here, but place the 238 00:15:44,290 --> 00:15:47,050 translation at an angle with respect to 239 00:15:47,050 --> 00:15:50,150 the reflection plane. 240 00:15:50,150 --> 00:15:53,870 So what we're going to do is to take a first motif that's 241 00:15:53,870 --> 00:15:59,460 right-handed, reflect it to a second one, which is 242 00:15:59,460 --> 00:16:00,790 left-handed. 243 00:16:00,790 --> 00:16:05,150 And then slide it along so that it sits up in the same 244 00:16:05,150 --> 00:16:09,820 position relative to this centered lattice point. 245 00:16:09,820 --> 00:16:12,580 So here's number three, translation leaves it 246 00:16:12,580 --> 00:16:13,830 left-handed. 247 00:16:16,140 --> 00:16:22,020 So I've taken the operation of reflection combined it with a 248 00:16:22,020 --> 00:16:27,540 translation that has a perpendicular component plus a 249 00:16:27,540 --> 00:16:30,950 parallel component, perpendicular and parallel 250 00:16:30,950 --> 00:16:34,220 meaning the orientation of these two components of the 251 00:16:34,220 --> 00:16:39,140 translation relative to the reflection operation. 252 00:16:39,140 --> 00:16:43,630 Anybody want to hazard a guess on how that first one is 253 00:16:43,630 --> 00:16:47,420 related to the third one? 254 00:16:47,420 --> 00:16:48,630 Number one is right-handed. 255 00:16:48,630 --> 00:16:50,710 Number two is left-handed, so it's got to 256 00:16:50,710 --> 00:16:51,960 be reflection, right? 257 00:16:56,490 --> 00:17:02,430 If I put a reflection plane in here, this one ought to be 258 00:17:02,430 --> 00:17:04,140 tilted like this. 259 00:17:04,140 --> 00:17:06,960 That's not going to do the job. 260 00:17:06,960 --> 00:17:07,910 Anybody got any idea? 261 00:17:07,910 --> 00:17:08,356 Yeah. 262 00:17:08,356 --> 00:17:10,589 AUDIENCE: What is that, T, T1 [INAUDIBLE]? 263 00:17:10,589 --> 00:17:10,939 PROFESSOR: OK. 264 00:17:10,939 --> 00:17:13,869 T parallel plus T perpendicular means it's a 265 00:17:13,869 --> 00:17:15,780 component of this translation. 266 00:17:15,780 --> 00:17:19,900 This is T. This has a part T perpendicular, and it has a 267 00:17:19,900 --> 00:17:23,562 part T parallel relative to the initial mirror plane. 268 00:17:33,230 --> 00:17:37,930 My friends, we have just stumbled headlong over a new 269 00:17:37,930 --> 00:17:41,910 type of symmetry operation, which we have discovered upon 270 00:17:41,910 --> 00:17:44,520 making this combination of mirror plane 271 00:17:44,520 --> 00:17:46,320 with a centered lattice. 272 00:17:46,320 --> 00:17:49,900 And it's come up to smack us rudely in the face even though 273 00:17:49,900 --> 00:17:51,850 we may not have been clever enough to 274 00:17:51,850 --> 00:17:53,920 think of it in advance. 275 00:17:53,920 --> 00:17:57,930 This is a new type of operation, and it is an 276 00:17:57,930 --> 00:18:01,760 operation that cannot be reduced to one of the simple 277 00:18:01,760 --> 00:18:04,140 operations that we've defined so far. 278 00:18:04,140 --> 00:18:08,050 You've got to take two steps to get from number one to 279 00:18:08,050 --> 00:18:10,600 number three. 280 00:18:10,600 --> 00:18:16,350 The way you can do it is to reflect along a locus that is 281 00:18:16,350 --> 00:18:20,000 one half of the way along the perpendicular part of the 282 00:18:20,000 --> 00:18:24,620 translation, exactly the same location as we found the 283 00:18:24,620 --> 00:18:28,660 symmetry plane positioned when the translation was normal to 284 00:18:28,660 --> 00:18:30,460 the first mirror plane. 285 00:18:30,460 --> 00:18:34,330 But yet we can't put the object down yet in the 286 00:18:34,330 --> 00:18:39,420 position that would be produced by translation, 287 00:18:39,420 --> 00:18:42,150 because our translation is inclined to the mirror plane. 288 00:18:42,150 --> 00:18:45,470 So I've got to take a second step. 289 00:18:45,470 --> 00:18:46,340 Reflect. 290 00:18:46,340 --> 00:18:47,900 Don't yet put it down yet. 291 00:18:47,900 --> 00:18:50,990 Before you put it down, slide it up parallel to the mirror 292 00:18:50,990 --> 00:18:55,710 plane by an amount that's equal to the part of the 293 00:18:55,710 --> 00:18:58,940 translation that is parallel to the mirror plane. 294 00:18:58,940 --> 00:19:01,690 So to summarize this before all these words get too 295 00:19:01,690 --> 00:19:06,300 confusing, I'm saying that a reflection operation combined 296 00:19:06,300 --> 00:19:09,900 with a general translation that has a perpendicular 297 00:19:09,900 --> 00:19:12,590 component in the parallel component relative to the 298 00:19:12,590 --> 00:19:16,550 locus of reflection is going to be a new operation, which 299 00:19:16,550 --> 00:19:20,150 I'll write as sigma tau, a reflection part and a 300 00:19:20,150 --> 00:19:24,630 translation tau that is parallel to the mirror plane, 301 00:19:24,630 --> 00:19:28,690 and tau is equal to the part of the translation that is 302 00:19:28,690 --> 00:19:32,580 parallel to the mirror plane. 303 00:19:37,090 --> 00:19:38,340 Astounding. 304 00:19:40,140 --> 00:19:43,670 This is a two-step operation that cannot be described in 305 00:19:43,670 --> 00:19:47,040 terms simpler than saying do two steps to do it. 306 00:19:47,040 --> 00:19:49,510 And we'll see as we go along, particularly into a 307 00:19:49,510 --> 00:19:52,310 three-dimensional space, that there are other two-step 308 00:19:52,310 --> 00:19:54,610 operations as well. 309 00:19:54,610 --> 00:19:59,520 Now you're all familiar with a pattern like this, because in 310 00:19:59,520 --> 00:20:02,810 very short order when New England's winter descends upon 311 00:20:02,810 --> 00:20:06,930 us, as you go slogging along from your room into the 312 00:20:06,930 --> 00:20:11,080 Institute, your footprints will make a pattern like that 313 00:20:11,080 --> 00:20:12,330 in the snow. 314 00:20:15,500 --> 00:20:17,280 Exactly what we've got here. 315 00:20:17,280 --> 00:20:20,110 Reflect across and slide. 316 00:20:20,110 --> 00:20:22,500 Reflect across and slide. 317 00:20:22,500 --> 00:20:26,250 Reflect across and slide, and this is the 318 00:20:26,250 --> 00:20:29,250 glide component tau. 319 00:20:29,250 --> 00:20:33,620 And this is an operation that is called a glide plane. 320 00:20:36,742 --> 00:20:39,490 And it's a new type of symmetry operation. 321 00:20:39,490 --> 00:20:41,790 It can only exist in a pattern, which has 322 00:20:41,790 --> 00:20:44,600 translational periodicity. 323 00:20:44,600 --> 00:20:47,200 And if we were not clever enough to invent it, we would 324 00:20:47,200 --> 00:20:51,370 see it as soon as we combined a mirror plane with a lattice 325 00:20:51,370 --> 00:20:54,340 that was non-primitive and had a translation parallel to it. 326 00:20:54,340 --> 00:20:57,810 So it's a very, very descriptive 327 00:20:57,810 --> 00:20:59,900 name, the glide plane. 328 00:20:59,900 --> 00:21:04,990 Reflect and glide, reflect and glide, reflect and glide. 329 00:21:04,990 --> 00:21:07,530 It sounds like something you'd be doing in the Arthur Murray 330 00:21:07,530 --> 00:21:09,500 dance studio, very melodious. 331 00:21:09,500 --> 00:21:12,430 Reflect and glide, reflect and glide. 332 00:21:12,430 --> 00:21:13,680 It's a nice operation. 333 00:21:19,730 --> 00:21:24,470 All right, so what has happened then when we add a 334 00:21:24,470 --> 00:21:29,320 mirror plane to a centered rectangular net is that we get 335 00:21:29,320 --> 00:21:35,880 a new operation coming in, something completely new. 336 00:21:35,880 --> 00:21:39,440 We've got a symbol to represent an individual 337 00:21:39,440 --> 00:21:42,350 operation, sigma the symbol for reflection with a 338 00:21:42,350 --> 00:21:44,670 subscript tau. 339 00:21:44,670 --> 00:21:50,550 And the pattern we've already drawn, but let's do it again 340 00:21:50,550 --> 00:21:53,670 in a tidy fashion. 341 00:21:53,670 --> 00:21:56,820 Exactly as advertised, the pair of objects related by 342 00:21:56,820 --> 00:22:00,020 reflection hung at every lattice point of-- holy 343 00:22:00,020 --> 00:22:00,610 mackerel, look at this. 344 00:22:00,610 --> 00:22:02,450 That would give you the willies. 345 00:22:02,450 --> 00:22:06,340 That's the nature of the motif. 346 00:22:06,340 --> 00:22:08,210 Get that out of there. 347 00:22:08,210 --> 00:22:10,175 Mirror planes going through the lattice points. 348 00:22:13,120 --> 00:22:15,460 Glide planes halfway in between. 349 00:22:19,390 --> 00:22:25,180 The mirror lines and the glide planes tell us how things on 350 00:22:25,180 --> 00:22:28,400 the right side, for example, of the lattice point are 351 00:22:28,400 --> 00:22:34,450 related to the motif of opposite chirality on the 352 00:22:34,450 --> 00:22:36,625 left-hand side of the centered lattice point. 353 00:22:39,320 --> 00:22:43,290 And this is a plane group that is called CM. 354 00:22:56,540 --> 00:22:57,540 AUDIENCE: Question. 355 00:22:57,540 --> 00:22:58,440 PROFESSOR: Yes, sir. 356 00:22:58,440 --> 00:23:01,179 AUDIENCE: How can we just define a new operation that's 357 00:23:01,179 --> 00:23:01,926 a two-step. 358 00:23:01,926 --> 00:23:04,914 Seems like we could do this forever, define two steps of 359 00:23:04,914 --> 00:23:07,260 any new operation, like you said, with two steps. 360 00:23:07,260 --> 00:23:08,200 PROFESSOR: That's a good question. 361 00:23:08,200 --> 00:23:11,330 We found this, because we tripped headlong over it. 362 00:23:11,330 --> 00:23:13,310 And we say OK, there it is. 363 00:23:13,310 --> 00:23:14,510 We've got to deal with it. 364 00:23:14,510 --> 00:23:15,200 But you're right. 365 00:23:15,200 --> 00:23:21,190 How do you know that rotating once reflecting, and then 366 00:23:21,190 --> 00:23:24,860 turning end over end three times is not a new operation 367 00:23:24,860 --> 00:23:28,310 that cannot be decomposed, is the word that's used for it, 368 00:23:28,310 --> 00:23:31,240 into something simpler? 369 00:23:31,240 --> 00:23:33,630 The answer is you've got to try it. 370 00:23:33,630 --> 00:23:37,810 If they're there, when we make these combinations of a 371 00:23:37,810 --> 00:23:41,240 symmetry operation, and now the symmetry operation can be 372 00:23:41,240 --> 00:23:44,090 a two-step symmetry operation, now that we've discovered 373 00:23:44,090 --> 00:23:47,190 that, combine that with lattice and with rotation and 374 00:23:47,190 --> 00:23:51,320 with reflection, and ask what is the relation between the 375 00:23:51,320 --> 00:23:54,760 motif at the beginning and the motif at the end. 376 00:23:54,760 --> 00:23:59,230 If you can't describe it any more simply other than a hop, 377 00:23:59,230 --> 00:24:02,590 skip and jump, you've got to introduce the hop, skip, and 378 00:24:02,590 --> 00:24:05,420 jump as an element that goes into the 379 00:24:05,420 --> 00:24:07,150 derivation of these groups. 380 00:24:07,150 --> 00:24:10,830 Now before you get concerned and fill out an add/drop card, 381 00:24:10,830 --> 00:24:15,100 I have to reassure you that there are a couple of two-step 382 00:24:15,100 --> 00:24:18,800 operations that we have yet to discover, but there are no 383 00:24:18,800 --> 00:24:20,710 three-step operations that are necessary. 384 00:24:23,686 --> 00:24:24,590 Whew. 385 00:24:24,590 --> 00:24:26,574 Feel better now, don't you? 386 00:24:26,574 --> 00:24:28,035 AUDIENCE: I was just going to say it's kind of arbitrary, 387 00:24:28,035 --> 00:24:30,957 because you have a and b, you're just applying the third 388 00:24:30,957 --> 00:24:31,444 [INAUDIBLE] 389 00:24:31,444 --> 00:24:35,840 T sub a and b, [INAUDIBLE] trivial multiplication tables. 390 00:24:35,840 --> 00:24:37,650 PROFESSOR: Well, actually, this is 391 00:24:37,650 --> 00:24:39,540 something that is distinct. 392 00:24:39,540 --> 00:24:42,710 I mean here is the group, and you cannot describe the 393 00:24:42,710 --> 00:24:47,200 relation between everything that is in this pattern, which 394 00:24:47,200 --> 00:24:50,350 was obtained simply by taking the operation of reflection-- 395 00:24:50,350 --> 00:24:53,060 we know how to peacefully coexist with that-- 396 00:24:53,060 --> 00:24:58,180 and placing that pair of objects at lattice point of a 397 00:24:58,180 --> 00:24:59,430 centered rectangular net. 398 00:25:01,916 --> 00:25:04,360 So that's nothing really freaky. 399 00:25:04,360 --> 00:25:08,430 I mean it's a straightforward addition, but if it's a group, 400 00:25:08,430 --> 00:25:11,650 you have to know when you combine all the operations 401 00:25:11,650 --> 00:25:15,490 pairwise that it'd be able to show that these operations are 402 00:25:15,490 --> 00:25:16,820 members of a group. 403 00:25:16,820 --> 00:25:19,560 And the answer in terms of the language of group theory, if 404 00:25:19,560 --> 00:25:22,770 you combine a reflection with a translation that has a 405 00:25:22,770 --> 00:25:26,420 component that is parallel to the reflection plane, then 406 00:25:26,420 --> 00:25:30,970 there is a new operation that comes up that has to be in the 407 00:25:30,970 --> 00:25:34,660 group multiplication table, and the operation has a 408 00:25:34,660 --> 00:25:38,910 reflection part and a translation part. 409 00:25:38,910 --> 00:25:40,100 AUDIENCE: I have a question. 410 00:25:40,100 --> 00:25:40,700 PROFESSOR: Yes. 411 00:25:40,700 --> 00:25:42,785 AUDIENCE: How do we go from the lower left to the upper 412 00:25:42,785 --> 00:25:44,630 right in one operation? 413 00:25:44,630 --> 00:25:47,350 PROFESSOR: The lower left to the upper right. 414 00:25:47,350 --> 00:25:51,670 AUDIENCE: No, the upper right diagonally. 415 00:25:51,670 --> 00:25:57,116 Across the diagonal, the upper right corner of the square. 416 00:25:57,116 --> 00:25:58,532 PROFESSOR: Upper right here? 417 00:25:58,532 --> 00:25:59,948 AUDIENCE: Yeah. 418 00:25:59,948 --> 00:26:01,943 And the left-handed guide below that. 419 00:26:01,943 --> 00:26:03,470 PROFESSOR: The left-handed guide below. 420 00:26:03,470 --> 00:26:04,220 AUDIENCE: No. 421 00:26:04,220 --> 00:26:06,090 Down there. 422 00:26:06,090 --> 00:26:07,272 PROFESSOR: Where is down there? 423 00:26:07,272 --> 00:26:08,565 AUDIENCE: The lowest edge of the-- 424 00:26:08,565 --> 00:26:09,430 PROFESSOR: Down here? 425 00:26:09,430 --> 00:26:09,820 AUDIENCE: Yeah. 426 00:26:09,820 --> 00:26:10,210 PROFESSOR: OK. 427 00:26:10,210 --> 00:26:11,655 From this one to this one? 428 00:26:11,655 --> 00:26:12,905 AUDIENCE: Yeah. 429 00:26:22,630 --> 00:26:23,410 PROFESSOR: OK. 430 00:26:23,410 --> 00:26:26,350 What I can say is-- 431 00:26:26,350 --> 00:26:29,670 may sound like I'm slipping off the hook too easily-- 432 00:26:29,670 --> 00:26:33,150 we said that we really only want to consider operations 433 00:26:33,150 --> 00:26:37,030 that terminate within the cell, and the way I get from 434 00:26:37,030 --> 00:26:42,430 here to here is to reflect and then translate up by the 435 00:26:42,430 --> 00:26:43,670 diagonal translation. 436 00:26:43,670 --> 00:26:46,950 So that's something that lies outside the cell. 437 00:26:46,950 --> 00:26:49,470 So I can always knock off an integral number of 438 00:26:49,470 --> 00:26:52,470 translations or add on an integral number of 439 00:26:52,470 --> 00:26:58,026 translations to any mapping transformation, modulo T. OK? 440 00:26:58,026 --> 00:26:59,994 AUDIENCE: But surely, if you've done, one operation 441 00:26:59,994 --> 00:27:02,460 then it's [INAUDIBLE]? 442 00:27:02,460 --> 00:27:03,383 PROFESSOR: From here to here? 443 00:27:03,383 --> 00:27:04,940 No, not necessarily. 444 00:27:04,940 --> 00:27:10,840 If I give you a very simple pattern and plane group P1, 445 00:27:10,840 --> 00:27:15,310 and you ask how do I get from this one here to this one that 446 00:27:15,310 --> 00:27:20,175 sits up here? 447 00:27:20,175 --> 00:27:24,150 If it's outside the cell, I've got to go translation that's 448 00:27:24,150 --> 00:27:24,940 outside the cell. 449 00:27:24,940 --> 00:27:26,770 But it's not any new translation or 450 00:27:26,770 --> 00:27:30,000 any new sort of operation. 451 00:27:30,000 --> 00:27:32,300 This is sort of in the same category. 452 00:27:32,300 --> 00:27:33,030 Yeah. 453 00:27:33,030 --> 00:27:34,950 AUDIENCE: You can make a glide plane in the 454 00:27:34,950 --> 00:27:36,970 center of the cell. 455 00:27:36,970 --> 00:27:38,850 PROFESSOR: Yeah, OK. 456 00:27:38,850 --> 00:27:40,510 Thank you. 457 00:27:40,510 --> 00:27:45,500 There is a glide operation that goes from here to here 458 00:27:45,500 --> 00:27:49,190 and then translates up by one full translation. 459 00:27:49,190 --> 00:27:52,650 But a glide operation that is an integral number of 460 00:27:52,650 --> 00:27:55,990 translations says that you're dealing with-- 461 00:27:55,990 --> 00:28:00,720 there's another object that is removed by a translation that 462 00:28:00,720 --> 00:28:01,870 is the same thing. 463 00:28:01,870 --> 00:28:05,620 So it is first and that simpler one that would be 464 00:28:05,620 --> 00:28:08,570 inside of the cell and would be only the unique sort of 465 00:28:08,570 --> 00:28:11,630 translation you need to consider. 466 00:28:11,630 --> 00:28:11,980 Thank you. 467 00:28:11,980 --> 00:28:13,020 That's a good question. 468 00:28:13,020 --> 00:28:15,720 That was a good answer to his question. 469 00:28:15,720 --> 00:28:17,260 In fact, anybody want to take over? 470 00:28:17,260 --> 00:28:18,127 I didn't do well on that one. 471 00:28:18,127 --> 00:28:19,192 Yeah. 472 00:28:19,192 --> 00:28:20,174 AUDIENCE: I was curious. 473 00:28:20,174 --> 00:28:22,629 Does the glide plane exist maybe in this case because 474 00:28:22,629 --> 00:28:25,084 we're not using a primitive cell? 475 00:28:25,084 --> 00:28:29,020 If you were to consider this [INAUDIBLE] as hexagonal? 476 00:28:29,020 --> 00:28:32,950 PROFESSOR: OK, let me answer that question by saying that 477 00:28:32,950 --> 00:28:36,370 that's our first encounter with it. 478 00:28:36,370 --> 00:28:41,230 But now, that it exists, that is a symmetry element, which 479 00:28:41,230 --> 00:28:46,960 we should consider adding to lattices in addition to pure 480 00:28:46,960 --> 00:28:48,600 reflection. 481 00:28:48,600 --> 00:28:53,540 So let me proceed now to do another plane group since we 482 00:28:53,540 --> 00:28:56,766 had discovered the operation of glide. 483 00:28:56,766 --> 00:29:06,060 And I will take a primitive rectangular lattice, and I 484 00:29:06,060 --> 00:29:11,565 will now add to the lattice point, not a mirror plane, but 485 00:29:11,565 --> 00:29:13,136 a glide plane. 486 00:29:13,136 --> 00:29:14,950 OK. 487 00:29:14,950 --> 00:29:18,740 The pattern is going to look like the pattern of glide, 488 00:29:18,740 --> 00:29:20,950 have things left and right on either side 489 00:29:20,950 --> 00:29:22,120 of the glide plane. 490 00:29:22,120 --> 00:29:23,370 The same thing here. 491 00:29:27,440 --> 00:29:30,700 And there is the pattern. 492 00:29:30,700 --> 00:29:32,502 This is a pattern that's called PG. 493 00:29:37,540 --> 00:29:43,880 And what I have to ask is what is the guide operation sigma 494 00:29:43,880 --> 00:29:47,790 tau combined with, let's say this translation T1. 495 00:29:50,330 --> 00:30:00,760 And the answer is that if I reproduce number one to number 496 00:30:00,760 --> 00:30:05,690 two of opposite chirality by the glide operation sigma tau 497 00:30:05,690 --> 00:30:13,980 and then follow that by T1 the first and the third will be 498 00:30:13,980 --> 00:30:19,590 related by a new glide operation sigma tau prime 499 00:30:19,590 --> 00:30:22,510 that's located at half perpendicular part of the 500 00:30:22,510 --> 00:30:25,180 translation away from the first. 501 00:30:25,180 --> 00:30:28,800 And that really is a generalization of this 502 00:30:28,800 --> 00:30:30,910 relation here. 503 00:30:30,910 --> 00:30:38,270 If we make the pure reflection operation a glide operation, 504 00:30:38,270 --> 00:30:42,390 combine it with a translation that is perpendicular to it, 505 00:30:42,390 --> 00:30:45,370 it turns out that the net result is a new glide 506 00:30:45,370 --> 00:30:48,290 operation, sigma tau prime. 507 00:30:48,290 --> 00:30:54,580 The two taus are equal, and this occurs at one half of the 508 00:30:54,580 --> 00:30:58,200 translation T perpendicular removed from the first. 509 00:30:58,200 --> 00:31:02,660 And the complete generalization would be to say 510 00:31:02,660 --> 00:31:06,500 if I have a translation that has a parallel part and a 511 00:31:06,500 --> 00:31:12,540 perpendicular part relative to the glide plane, what I will 512 00:31:12,540 --> 00:31:18,950 get is a new glide operation sigma prime that's located at 513 00:31:18,950 --> 00:31:21,870 1/2 of the perpendicular part of the translation from the 514 00:31:21,870 --> 00:31:25,190 first, and it will pick up a glide component that's equal 515 00:31:25,190 --> 00:31:28,490 to the original tau plus the part of the translation that's 516 00:31:28,490 --> 00:31:31,190 parallel to the glide plane. 517 00:31:31,190 --> 00:31:34,820 So this now is this theorem involving translation and 518 00:31:34,820 --> 00:31:37,760 reflection type operations in its most general form. 519 00:31:42,960 --> 00:31:43,330 OK. 520 00:31:43,330 --> 00:31:50,060 So this is a new group called CM, and it consists of pairs 521 00:31:50,060 --> 00:31:51,310 of objects. 522 00:31:54,740 --> 00:31:55,810 Sorry, we did that earlier. 523 00:31:55,810 --> 00:31:59,180 This is PG, pairs of objects related by a glide plane, hung 524 00:31:59,180 --> 00:32:03,940 at every lattice point of the primitive rectangular lattice. 525 00:32:03,940 --> 00:32:05,372 Is there a CG? 526 00:32:12,200 --> 00:32:17,900 Actually, there is not, and let me show you why, and then 527 00:32:17,900 --> 00:32:20,380 I think we're just about quitting time. 528 00:32:20,380 --> 00:32:25,070 If there is a centered lattice, and I hang a glide 529 00:32:25,070 --> 00:32:31,610 plane at the lattice points, we'll have glide planes in the 530 00:32:31,610 --> 00:32:39,140 locations of PG, the pattern will look like objects 531 00:32:39,140 --> 00:32:40,390 repeated by glide. 532 00:32:42,780 --> 00:32:46,670 At this lattice point, and now we're going to have another 533 00:32:46,670 --> 00:32:50,540 object hung at the centered lattice point, and it's going 534 00:32:50,540 --> 00:32:53,520 to be in positions like this. 535 00:32:53,520 --> 00:33:00,560 Now, you can see just by looking at the pattern that 536 00:33:00,560 --> 00:33:02,290 there is going to be-- 537 00:33:02,290 --> 00:33:05,220 whoops, what did I do here? 538 00:33:05,220 --> 00:33:07,900 I want to move this one up to here, and I want to move this 539 00:33:07,900 --> 00:33:09,380 one up to here. 540 00:33:09,380 --> 00:33:11,170 This one should be in this location. 541 00:33:16,000 --> 00:33:20,240 If I look at that pattern, what I've done is to create 542 00:33:20,240 --> 00:33:23,790 halfway along the-- 543 00:33:23,790 --> 00:33:26,530 quarter of the way along the translation a 544 00:33:26,530 --> 00:33:29,990 pure reflection operation. 545 00:33:29,990 --> 00:33:35,190 And I can find that using my general theorem. 546 00:33:35,190 --> 00:33:42,160 I have a glide operation, sigma tau. 547 00:33:42,160 --> 00:33:45,500 I combine with that the centered translation, which is 548 00:33:45,500 --> 00:33:50,520 1/2 of T1 plus 1/2 of T2. 549 00:33:54,680 --> 00:33:59,200 I've taken 1/2 of T1 plus 1/2 of T2, that's the centering 550 00:33:59,200 --> 00:34:03,846 translation, combined that with sigma tau. 551 00:34:03,846 --> 00:34:07,060 I'll deftly jump to the side, so the people against the wall 552 00:34:07,060 --> 00:34:09,110 can see it. 553 00:34:09,110 --> 00:34:12,650 I've taken a glide operation, combined it with this 554 00:34:12,650 --> 00:34:14,260 translation. 555 00:34:14,260 --> 00:34:20,679 This should be a new reflection type of operation 556 00:34:20,679 --> 00:34:24,810 that will be located at 1/2 of the perpendicular part of the 557 00:34:24,810 --> 00:34:32,219 translation, and that's at 1/2 of 1/2 of T1. 558 00:34:32,219 --> 00:34:37,219 And it will have a glide component tau prime, which is 559 00:34:37,219 --> 00:34:43,719 the original tau, which was 1/2 of T2. 560 00:34:43,719 --> 00:34:46,130 And to that we add the parallel part of the 561 00:34:46,130 --> 00:34:50,110 translation, and that is 1/2 of T2. 562 00:34:50,110 --> 00:34:55,510 So this is rewritten in slightly different form, is 563 00:34:55,510 --> 00:35:01,300 simply a new glide plane sigma prime, which has a glide 564 00:35:01,300 --> 00:35:05,620 component equal to the entire translation T2 at 565 00:35:05,620 --> 00:35:08,110 1/4 quarter of T1. 566 00:35:08,110 --> 00:35:12,200 And this is the same as a mirror plane at 1/4 T1. 567 00:35:16,250 --> 00:35:20,030 And this, if we compare it with CM, is 568 00:35:20,030 --> 00:35:22,600 exactly the same thing. 569 00:35:22,600 --> 00:35:36,400 Identical to CM with an origin shift of 1/4 T1. 570 00:35:36,400 --> 00:35:41,880 So there is no CG in what we picked up in terms of 571 00:35:41,880 --> 00:35:50,290 rectangular nets and symmetry planes is PM, PG, and CN. 572 00:35:50,290 --> 00:35:54,510 So there are three groups involving orthogonal nets in a 573 00:35:54,510 --> 00:35:55,760 single symmetry plane. 574 00:35:58,510 --> 00:36:03,870 That is a good place to wrap things up. 575 00:36:03,870 --> 00:36:08,120 And we'll next turn very, very quickly equipped with a set of 576 00:36:08,120 --> 00:36:11,240 notes, which summarizes the result, to what happens when 577 00:36:11,240 --> 00:36:16,600 we take two MM and put it into a rectangular net, and take 578 00:36:16,600 --> 00:36:20,900 two MM and put it into a centered rectangular net. 579 00:36:20,900 --> 00:36:23,130 And then things get interesting, because we've got 580 00:36:23,130 --> 00:36:23,720 two planes. 581 00:36:23,720 --> 00:36:25,330 We can make a both mirror planes. 582 00:36:25,330 --> 00:36:28,150 We can make them both glides, or make one a mirror plane and 583 00:36:28,150 --> 00:36:29,320 one a glide plane. 584 00:36:29,320 --> 00:36:31,110 So there are three possibilities with the 585 00:36:31,110 --> 00:36:34,880 addition of two MM to the net. 586 00:36:34,880 --> 00:36:36,850 More than enough for one day. 587 00:36:36,850 --> 00:36:39,330 It's Thursday. 588 00:36:39,330 --> 00:36:41,580 Take the rest of the week and weekend 589 00:36:41,580 --> 00:36:44,130 off by doing no symmetry. 590 00:36:44,130 --> 00:36:46,840 There's no assignment, and we'll have at 591 00:36:46,840 --> 00:36:48,090 it again next Tuesday.