1 00:00:07,400 --> 00:00:08,330 PROFESSOR: OK. 2 00:00:08,330 --> 00:00:14,100 Before we get started, I'd like to deal with a small 3 00:00:14,100 --> 00:00:17,960 matter of some unpleasantness. 4 00:00:17,960 --> 00:00:20,280 The class is sort of like a football game. 5 00:00:20,280 --> 00:00:23,030 When there's two minutes to go, you shoot off a pistol. 6 00:00:23,030 --> 00:00:25,330 But when there are two meetings to go until we have 7 00:00:25,330 --> 00:00:30,740 the quiz, we shoot off a pistol to wake you up. 8 00:00:30,740 --> 00:00:34,250 We're scheduled to have a quiz on October 6. 9 00:00:34,250 --> 00:00:37,680 And even though October seems far away when you're still in 10 00:00:37,680 --> 00:00:40,080 September, that is going to be a week 11 00:00:40,080 --> 00:00:41,920 from this coming Thursday. 12 00:00:41,920 --> 00:00:46,580 So if you have problems that you're working on, try to get 13 00:00:46,580 --> 00:00:49,560 them to me on Thursday, or just come slide them under my 14 00:00:49,560 --> 00:00:50,840 office door, if I'm not in. 15 00:00:50,840 --> 00:00:55,330 And I'll have them back for you on this coming Tuesday. 16 00:00:55,330 --> 00:00:58,450 The way the material has played out is that we're 17 00:00:58,450 --> 00:01:03,000 really at a nice, convenient juncture between one chunk of 18 00:01:03,000 --> 00:01:06,020 interconnected material moving on to another. 19 00:01:06,020 --> 00:01:08,340 So I think the first quiz we'll confine to 20 00:01:08,340 --> 00:01:10,025 two-dimensional symmetry. 21 00:01:10,025 --> 00:01:12,900 And beginning in about two minutes flat, we'll begin to 22 00:01:12,900 --> 00:01:14,110 move into-- 23 00:01:14,110 --> 00:01:16,830 take the first small steps, at any rate, into 24 00:01:16,830 --> 00:01:19,190 three-dimensional symmetries, which will be much more 25 00:01:19,190 --> 00:01:23,470 complicated in which we will not deal with the 26 00:01:23,470 --> 00:01:26,180 exhaustiveness that we have been able to afford the luxury 27 00:01:26,180 --> 00:01:29,230 of in two dimensions. 28 00:01:29,230 --> 00:01:32,400 Before I begin, let me-- does everybody remember their 29 00:01:32,400 --> 00:01:35,710 spherical trigonometry? 30 00:01:35,710 --> 00:01:39,850 Has anybody had spherical trigonometry? 31 00:01:39,850 --> 00:01:42,090 OK. 32 00:01:42,090 --> 00:01:45,950 What I will then do is give a short primer on some of the 33 00:01:45,950 --> 00:01:49,250 definitions and concepts in spherical trigonometry. 34 00:01:49,250 --> 00:01:53,960 And then we shall immediately use this to combine rotation 35 00:01:53,960 --> 00:01:57,310 axes in space. 36 00:01:57,310 --> 00:01:58,560 Pass those back. 37 00:02:02,738 --> 00:02:04,220 AUDIENCE: I had a quick question on this. 38 00:02:04,220 --> 00:02:05,580 PROFESSOR: Oh, sure, please. 39 00:02:05,580 --> 00:02:09,118 AUDIENCE: With the limiting possible-- 40 00:02:09,118 --> 00:02:09,601 PROFESSOR: Yep. 41 00:02:09,601 --> 00:02:12,016 AUDIENCE: --reflections and conditions, that does say they 42 00:02:12,016 --> 00:02:13,950 have to be figured out? 43 00:02:13,950 --> 00:02:16,380 PROFESSOR: Actually, that was a good question and something 44 00:02:16,380 --> 00:02:20,000 that we are not going to use at all in the symmetry tables. 45 00:02:20,000 --> 00:02:25,210 Somebody asked what about this notation in the far right-hand 46 00:02:25,210 --> 00:02:27,180 edge of all of the plane groups-- 47 00:02:27,180 --> 00:02:29,226 conditions limiting possible reflections. 48 00:02:33,140 --> 00:02:35,820 One usually doesn't put a two-dimensional crystal in an 49 00:02:35,820 --> 00:02:39,540 x-ray beam fairly often, although I suppose a thin film 50 00:02:39,540 --> 00:02:42,710 actually is almost a two-dimensional crystal. 51 00:02:42,710 --> 00:02:47,560 But you've probably all heard one way or another about the 52 00:02:47,560 --> 00:02:54,840 magic conditions relating the Miller indices of a plane that 53 00:02:54,840 --> 00:02:58,650 will require that the intensity diffracted from that 54 00:02:58,650 --> 00:03:01,910 set of planes is identically zero. 55 00:03:01,910 --> 00:03:06,980 And they're linear combinations of H, K, and L. 56 00:03:06,980 --> 00:03:13,100 And one of the rules is that if H plus K plus L is even or 57 00:03:13,100 --> 00:03:17,680 not, the intensity may be zero. 58 00:03:17,680 --> 00:03:21,380 These are rules for systematic absences. 59 00:03:21,380 --> 00:03:26,890 And the corresponding information is given for you 60 00:03:26,890 --> 00:03:29,930 here for these not terribly realistic real 61 00:03:29,930 --> 00:03:31,520 two-dimensional crystals. 62 00:03:31,520 --> 00:03:36,170 So, for example, if you turn to number seven, P2MG, it says 63 00:03:36,170 --> 00:03:40,320 conditions limiting possible reflections 64 00:03:40,320 --> 00:03:41,630 for the general position. 65 00:03:41,630 --> 00:03:44,310 For H and K, there's no condition. 66 00:03:44,310 --> 00:03:51,610 For H zero, H has to be even, if the reflection is to have 67 00:03:51,610 --> 00:03:54,040 non-zero intensity. 68 00:03:54,040 --> 00:04:02,070 And for the last two, for the general reflections, H, K, if 69 00:04:02,070 --> 00:04:08,060 there's an atom occupying the position either zero, zero or 70 00:04:08,060 --> 00:04:12,490 the position zero, one half, then for the general planes 71 00:04:12,490 --> 00:04:16,230 with indices H, K, you will see intensity only if H is 72 00:04:16,230 --> 00:04:18,339 even, as well-- same as the condition above. 73 00:04:21,470 --> 00:04:25,960 This is something that is not generally known that everybody 74 00:04:25,960 --> 00:04:29,340 knows-- that if the crystal is face-centered cubic, there is 75 00:04:29,340 --> 00:04:31,130 a pattern of absences. 76 00:04:31,130 --> 00:04:35,470 But there are additional absences if the atom is in a 77 00:04:35,470 --> 00:04:37,700 special position. 78 00:04:37,700 --> 00:04:40,760 And this can very often be used to advantage because you 79 00:04:40,760 --> 00:04:45,690 can single out certain classes of Miller indices for which 80 00:04:45,690 --> 00:04:49,570 one atom in the structure will not diffract or which another 81 00:04:49,570 --> 00:04:52,620 atom in the structure will not diffract. 82 00:04:52,620 --> 00:04:55,020 And that can be of great utility in 83 00:04:55,020 --> 00:04:57,080 unraveling a structure. 84 00:04:57,080 --> 00:05:00,050 We'll see some examples of this in useful form when we 85 00:05:00,050 --> 00:05:02,090 deal with three-dimensional space groups. 86 00:05:02,090 --> 00:05:05,510 And the corresponding sheets for the three-dimensional 87 00:05:05,510 --> 00:05:07,900 symmetries will be handed out to you-- some of 88 00:05:07,900 --> 00:05:09,150 them, not all of them. 89 00:05:11,170 --> 00:05:15,600 Let me take a little bit of time to remind you, if you've 90 00:05:15,600 --> 00:05:24,670 forgotten them, but to inform you of certain definitions and 91 00:05:24,670 --> 00:05:28,820 trigonometry in spherical geometry. 92 00:05:28,820 --> 00:05:34,220 Spherical trigonometry differs from plane geometry in that 93 00:05:34,220 --> 00:05:38,310 all of the action takes place on the surface of a sphere. 94 00:05:38,310 --> 00:05:41,190 And it'd be nice if I had a spherical blackboard. 95 00:05:41,190 --> 00:05:44,210 Actually there is one in the x-ray laboratory that I can 96 00:05:44,210 --> 00:05:46,710 draw things right on that spherical surface. 97 00:05:46,710 --> 00:05:49,550 But this is a sphere. 98 00:05:49,550 --> 00:05:51,935 We'll see directly that the radius of the 99 00:05:51,935 --> 00:05:53,840 sphere is not important. 100 00:05:53,840 --> 00:05:58,560 So we'll take that as a unity, which is a nice, even number. 101 00:05:58,560 --> 00:06:04,200 And as my dichotomy of the afternoon, if we pass a plane 102 00:06:04,200 --> 00:06:09,990 through that sphere, if the plane hits the sphere, it will 103 00:06:09,990 --> 00:06:12,970 intersect it in a circle. 104 00:06:12,970 --> 00:06:16,230 If the plane passes through the center of the sphere-- 105 00:06:16,230 --> 00:06:19,830 and we've assigned the radius of the sphere as unity, then 106 00:06:19,830 --> 00:06:23,515 this is a circle that's referred to a great circle. 107 00:06:26,940 --> 00:06:29,670 Sounds like a value judgment, but it's simply saying that's 108 00:06:29,670 --> 00:06:32,370 as large as the circle is going to get is when it passes 109 00:06:32,370 --> 00:06:34,540 through the center of the sphere., it would have unit 110 00:06:34,540 --> 00:06:36,085 radius, just as the sphere does. 111 00:06:36,085 --> 00:06:39,130 So if you take any other plane which intersects the sphere 112 00:06:39,130 --> 00:06:41,680 but doesn't pass through the center, it's going to have a 113 00:06:41,680 --> 00:06:42,590 smaller radius. 114 00:06:42,590 --> 00:06:44,325 And this is something that's called a small circle. 115 00:06:52,530 --> 00:06:56,180 OK, so if all of the action is going to take place on the 116 00:06:56,180 --> 00:07:01,410 surface of the sphere, and we have two points on the sphere, 117 00:07:01,410 --> 00:07:07,360 A and B, sitting on the surface of the sphere, how do 118 00:07:07,360 --> 00:07:11,204 we measure the separation of A and B? 119 00:07:11,204 --> 00:07:15,310 Well, if you think in terms of a normal three-dimensional 120 00:07:15,310 --> 00:07:17,740 person, you say, zonk, connect them by a line. 121 00:07:17,740 --> 00:07:19,770 And that's the distance between A and B. Now, you 122 00:07:19,770 --> 00:07:23,180 can't do that because all the action has to take place on 123 00:07:23,180 --> 00:07:25,510 the surface of the sphere. 124 00:07:25,510 --> 00:07:28,750 People who deal with spherical trigonometry all the time are 125 00:07:28,750 --> 00:07:30,360 airplane pilots. 126 00:07:30,360 --> 00:07:34,060 And if your pilot is going to take you from New York-- 127 00:07:34,060 --> 00:07:34,860 where would you like to go? 128 00:07:34,860 --> 00:07:35,400 Paris? 129 00:07:35,400 --> 00:07:36,210 That sounds like a nice place. 130 00:07:36,210 --> 00:07:40,560 But if you're going to go from New York to Paris, you don't 131 00:07:40,560 --> 00:07:42,890 plow your way through the intervening earth. 132 00:07:42,890 --> 00:07:46,650 You follow something that is at a constant radius out from 133 00:07:46,650 --> 00:07:50,200 the center the Earth, at a height of 5,000 feet above the 134 00:07:50,200 --> 00:07:53,320 surface of the Earth, an additional radius. 135 00:07:53,320 --> 00:07:56,670 So the way we'll define distance is to pass a great 136 00:07:56,670 --> 00:08:01,700 circle through A, B, pass a plane through A, B in the 137 00:08:01,700 --> 00:08:03,610 center of the sphere. 138 00:08:03,610 --> 00:08:08,040 And then we will define distance between A and B as 139 00:08:08,040 --> 00:08:12,580 the smaller of the two angles subtended at the 140 00:08:12,580 --> 00:08:15,340 center of the sphere. 141 00:08:15,340 --> 00:08:18,490 So this is a more reasonable looking great circle. 142 00:08:18,490 --> 00:08:21,850 If this is point A and this is point B, pass a plane through 143 00:08:21,850 --> 00:08:25,410 the center of the sphere, O, and through A and through B. 144 00:08:25,410 --> 00:08:28,740 And then we'll measure the length of the arc AB in terms 145 00:08:28,740 --> 00:08:31,830 of the angle alpha subtended at the center of the sphere. 146 00:08:34,679 --> 00:08:35,799 So it's a crazy notion. 147 00:08:35,799 --> 00:08:39,630 We're measuring distance in terms of an angle. 148 00:08:39,630 --> 00:08:43,740 And if that's an angle, we can take a trigonometric function 149 00:08:43,740 --> 00:08:46,740 of that angle, like sine or cosine. 150 00:08:46,740 --> 00:08:49,990 And that blows the mind that you can take trigonometric 151 00:08:49,990 --> 00:08:51,130 functions of a distance. 152 00:08:51,130 --> 00:08:51,700 But we can. 153 00:08:51,700 --> 00:08:55,700 We'll see it's going to be useful to us, too. 154 00:08:55,700 --> 00:08:57,290 And then I emphasize again, we'll take this 155 00:08:57,290 --> 00:08:58,780 as the smaller distance. 156 00:08:58,780 --> 00:09:02,400 It'll be 360 degrees minus alpha. 157 00:09:02,400 --> 00:09:04,260 That would be the long way around 158 00:09:04,260 --> 00:09:10,432 from A to B. All right. 159 00:09:10,432 --> 00:09:14,270 We've defined now how we will draw 160 00:09:14,270 --> 00:09:15,820 distances between two points. 161 00:09:15,820 --> 00:09:18,750 Suppose I have three points on the surface of the sphere-- 162 00:09:18,750 --> 00:09:28,290 A, B, and C. I can pass a great circle through A and B. 163 00:09:28,290 --> 00:09:30,470 I know how to do that. 164 00:09:30,470 --> 00:09:35,180 I can pass a great circle through A and C. I 165 00:09:35,180 --> 00:09:36,910 know how to do that. 166 00:09:36,910 --> 00:09:40,700 And I can pass a great circle through B and C. 167 00:09:40,700 --> 00:09:44,130 So now I have defined something that is referred to 168 00:09:44,130 --> 00:09:45,605 as a spherical triangle. 169 00:09:54,850 --> 00:09:58,200 We know how to measure the length of 170 00:09:58,200 --> 00:10:00,010 the spherical triangle. 171 00:10:00,010 --> 00:10:06,780 Let's call the arc opposite the point of intersection A as 172 00:10:06,780 --> 00:10:11,970 little a and the length of the arc opposite B as a distance 173 00:10:11,970 --> 00:10:15,320 and angles little b, and the distance from A 174 00:10:15,320 --> 00:10:18,400 to C as little c. 175 00:10:18,400 --> 00:10:21,660 But there's something in between these arcs that looks 176 00:10:21,660 --> 00:10:25,810 like an angle analogous to the angle in a planar triangle. 177 00:10:25,810 --> 00:10:29,520 And how can we define that? 178 00:10:29,520 --> 00:10:40,300 Well, the arc AB is defined by a plane, a great circle. 179 00:10:40,300 --> 00:10:46,090 The arc AC is similarly defined by a plane that passes 180 00:10:46,090 --> 00:10:49,090 through a, c in the center of the sphere. 181 00:10:49,090 --> 00:10:56,200 And what we will define as the spherical angle BAC is the 182 00:10:56,200 --> 00:11:02,120 angle between the great circles that define the two 183 00:11:02,120 --> 00:11:03,920 different arcs. 184 00:11:03,920 --> 00:11:09,530 So if there's one great circle that defines the arc from A to 185 00:11:09,530 --> 00:11:13,790 B and another plane that defines the arc from A to C, 186 00:11:13,790 --> 00:11:18,140 we'll define as the spherical angle between those two arcs 187 00:11:18,140 --> 00:11:23,390 the angle between the planes that define the great circles. 188 00:11:23,390 --> 00:11:26,840 So we're going to call this angle in here between these 189 00:11:26,840 --> 00:11:29,210 two planes as angle BAC. 190 00:11:41,960 --> 00:11:45,245 Another construct that is a useful one-- 191 00:11:49,720 --> 00:11:53,050 suppose I look at the plane that I've used to define a 192 00:11:53,050 --> 00:11:59,260 great circle and at the center of the sphere construct a line 193 00:11:59,260 --> 00:12:04,530 that is perpendicular to the great circle. 194 00:12:04,530 --> 00:12:07,120 And if I extend that line, sooner or later it's going to 195 00:12:07,120 --> 00:12:12,300 poke out through the surface of the sphere. 196 00:12:12,300 --> 00:12:20,480 And I will refer to this point as the pole of arc AB, or the 197 00:12:20,480 --> 00:12:24,190 pole of the great circle that we've used to define arc AB. 198 00:12:27,680 --> 00:12:35,010 So the North Pole is actually the pole of the great circle 199 00:12:35,010 --> 00:12:36,750 that defines the equator. 200 00:12:36,750 --> 00:12:38,180 And clearly there are two poles. 201 00:12:38,180 --> 00:12:39,800 There's one in either direction. 202 00:12:39,800 --> 00:12:44,505 So there's a North Pole and a South Pole to this arc AB and 203 00:12:44,505 --> 00:12:46,460 to the great circle that defines it. 204 00:12:57,942 --> 00:12:59,436 AUDIENCE: I have a question. 205 00:12:59,436 --> 00:13:02,922 Why couldn't you define the angle BAC as the angle 206 00:13:02,922 --> 00:13:04,172 [INAUDIBLE]? 207 00:13:09,894 --> 00:13:12,384 PROFESSOR: You want to make a tangent here? 208 00:13:12,384 --> 00:13:13,634 AUDIENCE: Yeah. 209 00:13:22,880 --> 00:13:26,796 PROFESSOR: I don't know if that's really defined. 210 00:13:26,796 --> 00:13:31,800 In other words, if I'm saying I want a line that is tangent 211 00:13:31,800 --> 00:13:34,844 to the sphere, it doesn't fix its orientation. 212 00:13:34,844 --> 00:13:36,460 AUDIENCE: In the plane of the great circle. 213 00:13:36,460 --> 00:13:36,760 PROFESSOR: OK. 214 00:13:36,760 --> 00:13:38,010 In the plane of the great circle. 215 00:13:44,730 --> 00:13:46,350 Suppose you could if you wanted to. 216 00:13:46,350 --> 00:13:50,210 There are trigonometric qualities to defining the 217 00:13:50,210 --> 00:13:53,010 angle in the way that we have. 218 00:13:53,010 --> 00:14:00,030 And the construct really is not something confined to the 219 00:14:00,030 --> 00:14:02,390 surface of the sphere, and everything else that 220 00:14:02,390 --> 00:14:06,270 we are doing is. 221 00:14:06,270 --> 00:14:06,780 OK? 222 00:14:06,780 --> 00:14:12,940 So it's sort of the non sequitur because we started 223 00:14:12,940 --> 00:14:16,100 out by saying that everything has to take place on the 224 00:14:16,100 --> 00:14:17,730 surface of the sphere. 225 00:14:17,730 --> 00:14:19,760 There's things that we would do in our three-dimensional 226 00:14:19,760 --> 00:14:23,100 world, like defining distances between points as the shortest 227 00:14:23,100 --> 00:14:27,980 straight line, that are ruled out in spherical trigonometry. 228 00:14:27,980 --> 00:14:32,160 And I think something similar could be 229 00:14:32,160 --> 00:14:35,340 levied at your proposal. 230 00:14:35,340 --> 00:14:41,302 And the answer is we just don't do it that way. 231 00:14:41,302 --> 00:14:42,552 That's the real answer. 232 00:14:46,040 --> 00:14:46,420 OK. 233 00:14:46,420 --> 00:14:54,690 If I have not boggled your mind so far, let me go a bit 234 00:14:54,690 --> 00:14:56,885 further with another useful construct. 235 00:15:01,930 --> 00:15:06,420 We can see how we can define a pole of a great circle or a 236 00:15:06,420 --> 00:15:11,050 pole on an arc that is a portion of a great circle. 237 00:15:11,050 --> 00:15:18,750 Let me take a spherical triangle, A, B, and C. And 238 00:15:18,750 --> 00:15:22,810 I've got three great circles now, which have formed those 239 00:15:22,810 --> 00:15:27,640 arcs that make up the sides of my triangle. 240 00:15:27,640 --> 00:15:33,620 Let me now find the pole of arc CB. 241 00:15:33,620 --> 00:15:35,900 And that means we're going to go out 90 degrees to that 242 00:15:35,900 --> 00:15:37,720 plane through the center of the sphere. 243 00:15:37,720 --> 00:15:40,390 And that's going to define some point that I'll call A 244 00:15:40,390 --> 00:15:45,050 prime, that is the pole of arc BC. 245 00:15:45,050 --> 00:15:48,260 And I'm going to do the same thing for the other arcs that 246 00:15:48,260 --> 00:15:52,700 are sides of my great circle of my spherical triangle. 247 00:15:52,700 --> 00:15:55,400 I'll find the pole of arc AC. 248 00:15:55,400 --> 00:16:00,070 And I'm going to label that point as B prime. 249 00:16:00,070 --> 00:16:02,630 And, finally, there'll be another pole that is 250 00:16:02,630 --> 00:16:04,480 the pole of arc AB. 251 00:16:04,480 --> 00:16:08,100 And that's going to define a point C prime. 252 00:16:08,100 --> 00:16:11,440 Now I've got three points, I can connect these together and 253 00:16:11,440 --> 00:16:13,998 make another spherical triangle. 254 00:16:13,998 --> 00:16:17,670 AUDIENCE: How do you know to determine where the pole is? 255 00:16:17,670 --> 00:16:19,700 PROFESSOR: If you think of it in three dimensions, I got 256 00:16:19,700 --> 00:16:20,670 three different arcs. 257 00:16:20,670 --> 00:16:24,020 And for each one of them I am drawing a perpendicular to the 258 00:16:24,020 --> 00:16:26,310 plane of that great circle and looking at the 259 00:16:26,310 --> 00:16:28,460 point where it emerges. 260 00:16:28,460 --> 00:16:28,720 OK. 261 00:16:28,720 --> 00:16:32,060 So now if there's another arc, there'll be another great 262 00:16:32,060 --> 00:16:33,940 circle coming around like this. 263 00:16:33,940 --> 00:16:35,660 And I look for the pole of it. 264 00:16:35,660 --> 00:16:38,950 And that would be another one of the corners. 265 00:16:38,950 --> 00:16:42,860 This thing that I've constructed is bizarre. 266 00:16:42,860 --> 00:16:44,200 But it's given a special name. 267 00:16:44,200 --> 00:16:45,615 This is called the polar triangle. 268 00:16:53,660 --> 00:16:56,185 And it has some useful properties. 269 00:17:04,890 --> 00:17:09,530 A property of the polar triangle is that the two 270 00:17:09,530 --> 00:17:15,569 triangles, A, B, and C, and A prime, B prime, and C prime 271 00:17:15,569 --> 00:17:18,540 are mutually polar. 272 00:17:18,540 --> 00:17:21,800 That is, if I use the spherical triangle ABC to 273 00:17:21,800 --> 00:17:24,095 define and locate the three points A 274 00:17:24,095 --> 00:17:25,560 prime, B prime, C prime-- 275 00:17:25,560 --> 00:17:30,985 now if I reverse the process and find the whole of arc A 276 00:17:30,985 --> 00:17:36,960 prime C prime, that turns out to be point B. 277 00:17:36,960 --> 00:17:41,120 And if I take the arc of A prime B prime, that turns-- 278 00:17:41,120 --> 00:17:45,070 I'm sorry-- take the arc of B prime C prime, that turns out 279 00:17:45,070 --> 00:17:50,470 to be point A. So the two triangles are mutually polar. 280 00:17:50,470 --> 00:17:53,190 The polar triangle of the polar triangle is the triangle 281 00:17:53,190 --> 00:17:54,340 that we started with. 282 00:17:54,340 --> 00:17:54,810 Yeah? 283 00:17:54,810 --> 00:17:56,220 AUDIENCE: I guess I missed that. 284 00:17:56,220 --> 00:17:58,570 So if you take B prime through C prime, then 285 00:17:58,570 --> 00:17:59,510 all of that is going-- 286 00:17:59,510 --> 00:18:05,002 PROFESSOR: Yeah, I'm saying that the pole of this arc, A 287 00:18:05,002 --> 00:18:08,770 prime C prime, is this point here. 288 00:18:08,770 --> 00:18:18,450 And the way I can show that is to say that we got B by 289 00:18:18,450 --> 00:18:21,150 looking at the pole-- 290 00:18:21,150 --> 00:18:22,000 I'm sorry-- 291 00:18:22,000 --> 00:18:26,510 I got C by looking at the pole of the arc AB. 292 00:18:26,510 --> 00:18:33,840 So B is 90 degrees away from C prime. 293 00:18:33,840 --> 00:18:43,080 I found point A prime by finding the pole of arc CB. 294 00:18:43,080 --> 00:18:47,440 So B is 90 degrees from any point on that arc. 295 00:18:47,440 --> 00:18:51,140 So it's 90 degrees away from A. So B is 90 296 00:18:51,140 --> 00:18:52,200 degrees from A prime. 297 00:18:52,200 --> 00:18:54,040 B is 90 degrees C prime. 298 00:18:54,040 --> 00:19:00,290 And, therefore, it has to be the pole of that arc. 299 00:19:00,290 --> 00:19:01,510 Now that was a little too quick. 300 00:19:01,510 --> 00:19:02,740 That's written down in the notes. 301 00:19:02,740 --> 00:19:03,990 And that's why I wrote them out. 302 00:19:09,030 --> 00:19:12,310 One final thing and then we can put circle trigonometry to 303 00:19:12,310 --> 00:19:14,690 one side, and this is something 304 00:19:14,690 --> 00:19:18,490 that is not all obvious. 305 00:19:18,490 --> 00:19:26,500 If we look at a spherical triangle and simultaneously 306 00:19:26,500 --> 00:19:27,540 the polar triangle-- 307 00:19:27,540 --> 00:19:31,290 so let's say this is ABC. 308 00:19:31,290 --> 00:19:38,660 And here is the polar triangle A prime, C prime, B prime. 309 00:19:45,527 --> 00:19:53,170 It turns out that the spherical angle in one circle 310 00:19:53,170 --> 00:20:00,380 triangle and the length of the arc opposite it, namely this 311 00:20:00,380 --> 00:20:04,665 arc B prime C prime, are complementary-- 312 00:20:10,040 --> 00:20:13,300 supplementary, not complementary. 313 00:20:13,300 --> 00:20:21,900 And the way one would do that is to say that the measure of 314 00:20:21,900 --> 00:20:25,160 alpha is the length of this arc here. 315 00:20:27,710 --> 00:20:36,700 And this total side, B prime C prime, is equal to this arc 316 00:20:36,700 --> 00:20:41,490 plus this arc minus this length. 317 00:20:41,490 --> 00:20:43,910 And these two arcs are 90 degrees. 318 00:20:43,910 --> 00:20:47,820 So let me do as I've done in the notes. 319 00:20:47,820 --> 00:20:50,270 Let me call this point P prime. 320 00:20:50,270 --> 00:20:53,860 And I'll call this point Q prime. 321 00:20:53,860 --> 00:21:05,340 So my argument, it says that B prime is the pole of arc AC. 322 00:21:05,340 --> 00:21:13,170 And, therefore, B prime Q prime equals 323 00:21:13,170 --> 00:21:15,660 90 degrees in length. 324 00:21:15,660 --> 00:21:23,810 And then I would say that C prime is the pole of arc AB. 325 00:21:23,810 --> 00:21:30,710 And, therefore, the distance C prime P prime is also exactly 326 00:21:30,710 --> 00:21:32,670 90 degrees. 327 00:21:32,670 --> 00:21:39,940 And that says that B prime Q prime plus C prime P prime-- 328 00:21:39,940 --> 00:21:44,530 if I add those two together, it has to be 180 degrees. 329 00:21:44,530 --> 00:21:54,880 But I can write B prime C prime as B prime P prime plus 330 00:21:54,880 --> 00:22:03,020 P prime Q prime plus Q prime C prime plus the 331 00:22:03,020 --> 00:22:04,870 side P prime Q prime. 332 00:22:08,620 --> 00:22:10,080 And that's 180 degrees. 333 00:22:14,560 --> 00:22:17,390 But these three things that I've lumped together here are 334 00:22:17,390 --> 00:22:20,780 exactly the same as the length of the spherical polar 335 00:22:20,780 --> 00:22:22,280 triangle A prime. 336 00:22:22,280 --> 00:22:26,170 So what we've shown then is that A prime plus alpha is 337 00:22:26,170 --> 00:22:30,580 equal to 180 degrees-- 338 00:22:30,580 --> 00:22:31,780 QED. 339 00:22:31,780 --> 00:22:38,100 So this angle plus the side of the polar triangle add up to 340 00:22:38,100 --> 00:22:39,350 180 degrees. 341 00:22:43,400 --> 00:22:45,230 And that is not obvious at all. 342 00:22:50,350 --> 00:22:51,430 One final relation-- 343 00:22:51,430 --> 00:22:55,635 and this I will simply hand to you on a platter. 344 00:22:55,635 --> 00:22:57,570 I'm not about to derive it. 345 00:23:00,500 --> 00:23:05,140 Sides and angles in planar geometry are related. 346 00:23:05,140 --> 00:23:09,130 And there's a particularly useful relation in plane 347 00:23:09,130 --> 00:23:11,340 geometry that's called the Law of Cosines. 348 00:23:15,440 --> 00:23:16,800 So this is in plane geometry. 349 00:23:22,040 --> 00:23:27,280 And if you have a triangle that has sides a, b, c-- 350 00:23:27,280 --> 00:23:29,300 a general oblique triangle-- 351 00:23:29,300 --> 00:23:35,030 and it has angles A, B, and C, the Law of Cosines says that 352 00:23:35,030 --> 00:23:40,410 the side A is determined by c and b and the 353 00:23:40,410 --> 00:23:41,900 angle between them. 354 00:23:41,900 --> 00:23:42,630 And that's clear. 355 00:23:42,630 --> 00:23:46,440 If I specify this length, specify this length, specify 356 00:23:46,440 --> 00:23:49,960 that angle, things set up like a bowl of supercooled jello. 357 00:23:49,960 --> 00:23:52,830 And the triangle's completely specified. 358 00:23:52,830 --> 00:23:56,565 So a squared in the Law of Cosines is b squared plus c 359 00:23:56,565 --> 00:24:03,880 squared minus 2bc times the cosine of angle A. 360 00:24:03,880 --> 00:24:08,670 In a spherical triangle there is a similar sort of 361 00:24:08,670 --> 00:24:10,030 constraint. 362 00:24:10,030 --> 00:24:14,580 If we have a spherical triangle with sides a, b, and 363 00:24:14,580 --> 00:24:19,750 c, and spherical angles capital A, capital B, capital 364 00:24:19,750 --> 00:24:23,640 C, in the same way as specifying the spherical angle 365 00:24:23,640 --> 00:24:28,650 A and the lengths of the two sides c and b, specifies and 366 00:24:28,650 --> 00:24:33,080 fixes the spherical triangle entirely. 367 00:24:33,080 --> 00:24:36,810 This side must be determined by the length of side c, the 368 00:24:36,810 --> 00:24:40,460 length of side b, and the angle between them. 369 00:24:40,460 --> 00:24:43,960 And that, since everything is in terms of angles, is 370 00:24:43,960 --> 00:24:46,160 something that doesn't involve squares. 371 00:24:46,160 --> 00:24:49,510 It involves totally trigonometric expressions. 372 00:24:49,510 --> 00:24:53,490 And it turns out the cosine of this missing side a is given 373 00:24:53,490 --> 00:24:57,475 by the product of the cosines of the two other sides. 374 00:24:57,475 --> 00:25:00,680 So as I said, you can take a trigonometric function of a 375 00:25:00,680 --> 00:25:05,260 length, which sounds like an oxymoron. 376 00:25:05,260 --> 00:25:09,820 And it's the product of the cosines of the two known sides 377 00:25:09,820 --> 00:25:15,380 and times the sine of b sine of c times the cosine of the 378 00:25:15,380 --> 00:25:18,090 spherical angle A. And that is also 379 00:25:18,090 --> 00:25:19,340 called the Law of Cosines. 380 00:25:24,400 --> 00:25:28,490 And this is the corresponding case in spherical geometry. 381 00:25:34,130 --> 00:25:34,430 OK. 382 00:25:34,430 --> 00:25:35,740 So there's some machinery-- 383 00:25:35,740 --> 00:25:36,674 yes, sir? 384 00:25:36,674 --> 00:25:39,009 AUDIENCE: What's the difference between the sines 385 00:25:39,009 --> 00:25:41,811 of lowercase a and-- 386 00:25:41,811 --> 00:25:44,850 PROFESSOR: OK, the angles are the capital letters. 387 00:25:44,850 --> 00:25:47,230 This would be the angle between the great circles that 388 00:25:47,230 --> 00:25:47,740 defines the-- 389 00:25:47,740 --> 00:25:51,065 AUDIENCE: Since the radius is one, there's no difference 390 00:25:51,065 --> 00:25:54,446 between the angles and the-- 391 00:25:54,446 --> 00:25:56,378 [INAUDIBLE]? 392 00:25:56,378 --> 00:25:59,200 PROFESSOR: No. 393 00:25:59,200 --> 00:26:02,910 This angle is something, for example, we can choose. 394 00:26:02,910 --> 00:26:06,270 And depending on how long we want this arc to be, we can 395 00:26:06,270 --> 00:26:09,270 put the arc BC anywhere we like. 396 00:26:09,270 --> 00:26:10,722 AUDIENCE: Does that mean your radius [INAUDIBLE]? 397 00:26:14,600 --> 00:26:15,190 PROFESSOR: No. 398 00:26:15,190 --> 00:26:19,000 This would be, say, two points of the spherical triangle. 399 00:26:19,000 --> 00:26:22,120 Now we can pick any third point on the surface of the 400 00:26:22,120 --> 00:26:25,540 sphere, connect that with great circles, and here is a 401 00:26:25,540 --> 00:26:26,790 spherical triangle. 402 00:26:29,264 --> 00:26:29,710 OK? 403 00:26:29,710 --> 00:26:31,130 So I see. 404 00:26:31,130 --> 00:26:35,290 I think I see what your problem is. 405 00:26:35,290 --> 00:26:37,610 Here are the two planes. 406 00:26:37,610 --> 00:26:41,870 We define the angle of the spherical triangle as the 407 00:26:41,870 --> 00:26:44,940 angle subtended buy the great circle. 408 00:26:44,940 --> 00:26:48,590 So this is the definition of A. 409 00:26:48,590 --> 00:26:51,940 But now the other two points on the spherical triangle can 410 00:26:51,940 --> 00:26:54,060 be any point on these great circles. 411 00:26:54,060 --> 00:26:57,750 So this can be point B, and this can be point C. And my 412 00:26:57,750 --> 00:27:01,340 spherical triangle can be something like this. 413 00:27:01,340 --> 00:27:07,630 So the arc that defines the spherical angle A is a value 414 00:27:07,630 --> 00:27:12,240 that is independent from the length of the arc AC. 415 00:27:12,240 --> 00:27:15,910 That would be what is subtended at the 416 00:27:15,910 --> 00:27:17,160 center of the sphere. 417 00:27:25,610 --> 00:27:30,050 I'm going to have time just to set the stage for how we're 418 00:27:30,050 --> 00:27:32,336 going to use these relations. 419 00:27:35,200 --> 00:27:38,900 And the problem that I would like to raise and then apply 420 00:27:38,900 --> 00:27:46,720 spherical trigonometry to is the question if I go into 421 00:27:46,720 --> 00:27:54,990 three-dimensional space, there is no longer any requirement 422 00:27:54,990 --> 00:27:58,340 that rotation axes be all 423 00:27:58,340 --> 00:28:00,170 parallel to the same direction. 424 00:28:00,170 --> 00:28:03,120 In two dimensions the rotation points were really-- 425 00:28:03,120 --> 00:28:06,420 could be viewed as axes that were always perpendicular to 426 00:28:06,420 --> 00:28:10,140 the plane of the blackboard, the plane of the plane group. 427 00:28:10,140 --> 00:28:16,250 But now when I'm dealing with three-dimensional spaces, this 428 00:28:16,250 --> 00:28:20,420 could be the operation A alpha. 429 00:28:20,420 --> 00:28:23,220 And there is no reason whatsoever why we should not 430 00:28:23,220 --> 00:28:26,940 try to combine with this first operation a 431 00:28:26,940 --> 00:28:31,790 second rotation B beta-- 432 00:28:31,790 --> 00:28:35,090 a rotation through an angle beta about this axis B and a 433 00:28:35,090 --> 00:28:40,060 rotation of angle alpha through this axis A. If we're 434 00:28:40,060 --> 00:28:43,690 going to come up with a crystallographic combination, 435 00:28:43,690 --> 00:28:46,970 the angles alpha and beta have to be restricted to the 436 00:28:46,970 --> 00:28:51,410 angular rotations of either a onefold, a twofold, a 437 00:28:51,410 --> 00:28:54,420 threefold, a fourfold, or a sixfold axis, if the 438 00:28:54,420 --> 00:28:56,550 combination is going to be crystallographic. 439 00:28:56,550 --> 00:28:57,012 Yes, sir? 440 00:28:57,012 --> 00:28:59,322 AUDIENCE: You're just rotating around those lines? 441 00:28:59,322 --> 00:29:00,750 PROFESSOR: I'm rotating around those lines, yeah. 442 00:29:00,750 --> 00:29:05,200 So I'm saying that we're going to rotate an angle alpha about 443 00:29:05,200 --> 00:29:06,930 axis A. 444 00:29:06,930 --> 00:29:09,480 And now what I'm going to raise as 445 00:29:09,480 --> 00:29:11,100 a rhetorical question-- 446 00:29:11,100 --> 00:29:16,550 what is the rotation A alpha followed by B beta? 447 00:29:16,550 --> 00:29:23,440 So I rotate through the angle alpha about A. So here's my 448 00:29:23,440 --> 00:29:27,920 first motif, right-handed. 449 00:29:27,920 --> 00:29:31,560 Then I'll rotate alpha degrees. 450 00:29:31,560 --> 00:29:33,290 And this here's number two. 451 00:29:33,290 --> 00:29:36,380 And that will stay right-handed. 452 00:29:36,380 --> 00:29:40,840 Now, I will place on axis B the two constraints. 453 00:29:40,840 --> 00:29:44,010 These have to be crystallographic rotation 454 00:29:44,010 --> 00:29:52,380 angles, namely 360, 180, 120, 90, or 60. 455 00:29:52,380 --> 00:29:55,330 And I'll also, since I would like to obtain point group 456 00:29:55,330 --> 00:30:00,450 symmetries initially, I will require that axis A and axis B 457 00:30:00,450 --> 00:30:03,620 intersect at some point. 458 00:30:03,620 --> 00:30:07,090 And one of the variables in the combination will be this 459 00:30:07,090 --> 00:30:10,000 angle between the two rotation axes. 460 00:30:10,000 --> 00:30:14,660 So let's complete our combination of operations. 461 00:30:14,660 --> 00:30:17,490 I'll rotate from one to two by A alpha. 462 00:30:17,490 --> 00:30:19,560 If the first one is right-handed, the second motif 463 00:30:19,560 --> 00:30:21,310 is right-handed, as well. 464 00:30:21,310 --> 00:30:27,840 And then I will rotate beta degrees about B. And here will 465 00:30:27,840 --> 00:30:31,680 sit number three. 466 00:30:31,680 --> 00:30:32,930 And it will stay right-handed. 467 00:30:35,490 --> 00:30:41,190 So, again, the $64 question that we raise periodically-- 468 00:30:41,190 --> 00:30:46,020 what operation is the net effect of two successive 469 00:30:46,020 --> 00:30:48,755 rotations about a point of intersection? 470 00:30:51,984 --> 00:30:53,234 [INTERPOSING VOICES] 471 00:30:55,337 --> 00:30:56,290 PROFESSOR: Lots of opinions. 472 00:30:56,290 --> 00:30:57,230 Let's sort them out. 473 00:30:57,230 --> 00:30:58,980 I heard translation. 474 00:30:58,980 --> 00:31:02,720 Well, let's put down what it could be. 475 00:31:02,720 --> 00:31:07,720 We know that only translation and rotation leaves the 476 00:31:07,720 --> 00:31:09,550 chirality of the motif unchanged. 477 00:31:09,550 --> 00:31:12,263 So it's got to be one or the other. 478 00:31:12,263 --> 00:31:14,187 AUDIENCE: Since there's no reason for the general 479 00:31:14,187 --> 00:31:16,592 orientation of the two to say it has to be rotation around 480 00:31:16,592 --> 00:31:18,035 the third axis, so the question is 481 00:31:18,035 --> 00:31:19,385 what angle and what-- 482 00:31:19,385 --> 00:31:19,760 PROFESSOR: OK. 483 00:31:19,760 --> 00:31:21,790 That is exactly the problem. 484 00:31:21,790 --> 00:31:25,080 Now that we know the problem, we can go home early because 485 00:31:25,080 --> 00:31:28,380 we know what we're going to do next time. 486 00:31:28,380 --> 00:31:32,030 Well, let me expand a little bit. 487 00:31:32,030 --> 00:31:35,080 It can't be translation because clearly the separation 488 00:31:35,080 --> 00:31:38,030 of number one and number three depend on exactly where I 489 00:31:38,030 --> 00:31:39,230 place the first one. 490 00:31:39,230 --> 00:31:41,350 If I place it a little further out from A, then it's going to 491 00:31:41,350 --> 00:31:42,620 rotate to here. 492 00:31:42,620 --> 00:31:44,230 And then B is going to swing it off to 493 00:31:44,230 --> 00:31:45,960 some different location. 494 00:31:45,960 --> 00:31:49,160 So it can't be translation. 495 00:31:49,160 --> 00:31:52,220 And I don't think these guys, if I rotate this way and then 496 00:31:52,220 --> 00:31:54,600 I rotate this way, are going to be parallel to one another. 497 00:31:54,600 --> 00:31:56,200 I doubt that very much. 498 00:31:56,200 --> 00:31:57,450 So it's got to be a rotation. 499 00:32:03,250 --> 00:32:08,500 So without knowing how to find it, let's say that we can get 500 00:32:08,500 --> 00:32:12,520 from number one to number three in one shot through 501 00:32:12,520 --> 00:32:20,340 rotation of an angle gamma about some third axis, C. So 502 00:32:20,340 --> 00:32:23,575 the answer, in general, without being specific, is A 503 00:32:23,575 --> 00:32:26,650 alpha followed B beta has got to be equal to a third 504 00:32:26,650 --> 00:32:30,980 rotation, C, about a direction that has the same point of 505 00:32:30,980 --> 00:32:32,870 intersection with the first two axes. 506 00:32:36,880 --> 00:32:40,163 Now we've got some really, really tough constraints. 507 00:32:43,140 --> 00:32:46,860 Alpha is restricted to one of five values. 508 00:32:46,860 --> 00:32:49,785 Beta is restricted to one of five values. 509 00:32:53,470 --> 00:32:57,290 The third rotation, gamma, jolly well better be a 510 00:32:57,290 --> 00:33:03,360 crystallographic rotation and not something that is not a 511 00:33:03,360 --> 00:33:04,720 sub-multiple of 2 pi. 512 00:33:04,720 --> 00:33:07,730 And even if it is a sub-multiple of 2 pi, it has 513 00:33:07,730 --> 00:33:13,830 to be either 0 degrees, 120, so on. 514 00:33:13,830 --> 00:33:14,580 It has to be one of the 515 00:33:14,580 --> 00:33:16,530 crystallographic rotation angles. 516 00:33:16,530 --> 00:33:20,690 So what sort of relation can we get that would give us, 517 00:33:20,690 --> 00:33:26,180 first, the value of gamma in terms of alpha, beta, and the 518 00:33:26,180 --> 00:33:29,560 angle at which we combine them? 519 00:33:29,560 --> 00:33:34,890 So taking A and B as the five crystallographic rotation axes 520 00:33:34,890 --> 00:33:40,460 two at a time, we want to put them together, if we can, such 521 00:33:40,460 --> 00:33:45,630 that the angle makes the third rotation axis also be 522 00:33:45,630 --> 00:33:46,820 crystallographic. 523 00:33:46,820 --> 00:33:49,580 And then we would have to find its location. 524 00:33:49,580 --> 00:33:52,810 It looks like an impossible constraint . 525 00:33:52,810 --> 00:33:56,010 It looks absolutely impossible to do. 526 00:33:56,010 --> 00:34:00,280 We've got to put this first in quantitative form and then 527 00:34:00,280 --> 00:34:04,400 simply put in the values for alpha, beta, and gamma and 528 00:34:04,400 --> 00:34:07,360 find the angle that they have to be combined on to make 529 00:34:07,360 --> 00:34:11,010 this, if possible, be one of the crystallographics 530 00:34:11,010 --> 00:34:12,260 sub-multiples. 531 00:34:13,929 --> 00:34:16,830 That is not an easy problem to formulate. 532 00:34:16,830 --> 00:34:20,449 And, as I said a couple of times ago, the geometry that 533 00:34:20,449 --> 00:34:25,219 is the basis of this derivation was originally 534 00:34:25,219 --> 00:34:27,889 proposed by Euler. 535 00:34:27,889 --> 00:34:30,659 And it's known as Euler's Construction. 536 00:34:30,659 --> 00:34:35,090 I will have for you next time my own set of notes on this. 537 00:34:35,090 --> 00:34:37,010 We have finished with two-dimensional 538 00:34:37,010 --> 00:34:37,780 crystallography. 539 00:34:37,780 --> 00:34:41,850 So we are back to Buerger's book again. 540 00:34:41,850 --> 00:34:44,260 We had that little interlude. 541 00:34:44,260 --> 00:34:46,659 Buerger deals with Euler's Construction. 542 00:34:46,659 --> 00:34:49,850 But I don't think he's at his best in 543 00:34:49,850 --> 00:34:51,380 this particular section. 544 00:34:51,380 --> 00:34:53,300 So we'll take it a little more slowly. 545 00:34:53,300 --> 00:34:55,820 And next time we'll get around to deriving Euler's 546 00:34:55,820 --> 00:34:57,070 Construction.