1 00:00:07,275 --> 00:00:09,864 PROFESSOR: All right, I feel almost as though I should 2 00:00:09,864 --> 00:00:11,265 introduce myself all over again. 3 00:00:11,265 --> 00:00:14,070 It's been a week and a half since we had a lecture. 4 00:00:14,070 --> 00:00:18,938 So let me begin by reminding you of what we were doing. 5 00:00:18,938 --> 00:00:21,842 We had derived all of the plane groups, and for better 6 00:00:21,842 --> 00:00:24,262 or worse had put them behind us. 7 00:00:24,262 --> 00:00:27,488 And then we moved into three dimensions, where things get a 8 00:00:27,488 --> 00:00:31,125 lot more involved and a lot more complicated, and to say 9 00:00:31,125 --> 00:00:33,198 the least, a lot more numerous. 10 00:00:33,198 --> 00:00:38,654 And the first question we asked was to say, when we're 11 00:00:38,654 --> 00:00:43,118 in a three-dimensional space, we can combine a first 12 00:00:43,118 --> 00:00:46,838 rotation operation with a second rotation operation, B 13 00:00:46,838 --> 00:00:51,054 beta If we're to begin by deriving point groups-- that 14 00:00:51,054 --> 00:00:53,534 is to say, the least [INAUDIBLE] point in this 15 00:00:53,534 --> 00:00:56,014 three-dimensional space is not going to move. 16 00:00:56,014 --> 00:00:57,006 [INAUDIBLE] 17 00:00:57,006 --> 00:00:59,486 two axes to intersect a point. 18 00:00:59,486 --> 00:01:01,966 For a space group, they could be parallel to one another. 19 00:01:01,966 --> 00:01:04,694 But that's gonna be an infinite set of symmetry 20 00:01:04,694 --> 00:01:07,422 elements and operations that extends through all space. 21 00:01:07,422 --> 00:01:10,398 Then we asked the question-- rhetorically, because you knew 22 00:01:10,398 --> 00:01:11,886 I was going to answer-- 23 00:01:11,886 --> 00:01:18,606 what is the net result of a sequence of rotating from a 24 00:01:18,606 --> 00:01:23,070 first object to a second, and then picking up the second and 25 00:01:23,070 --> 00:01:26,720 rotating it to an angle beta about second axis. 26 00:01:26,720 --> 00:01:30,150 Begin the third one here. 27 00:01:30,150 --> 00:01:31,832 What is the net effect? 28 00:01:31,832 --> 00:01:34,292 And again, we could do that by the process of elimination. 29 00:01:34,292 --> 00:01:38,720 It has to be either translation or another 30 00:01:38,720 --> 00:01:42,410 rotation, because these are the only two generic sorts of 31 00:01:42,410 --> 00:01:45,116 operations which leaves the chirality unchanged. 32 00:01:45,116 --> 00:01:48,560 And I think I convinced you that indeed there was some 33 00:01:48,560 --> 00:01:52,530 third axis, C, which rotated directly from the first to the 34 00:01:52,530 --> 00:01:57,300 third by some different angle, gamma. 35 00:01:57,300 --> 00:01:59,687 So that is the consequence of combining the first two 36 00:01:59,687 --> 00:02:01,683 rotation axes. 37 00:02:01,683 --> 00:02:10,166 What they would anticipate is that the location and also the 38 00:02:10,166 --> 00:02:14,158 value of the rotation [INAUDIBLE] depends on alpha 39 00:02:14,158 --> 00:02:16,653 and beta, and also [INAUDIBLE] 40 00:02:16,653 --> 00:02:18,150 at which we combine them. 41 00:02:18,150 --> 00:02:20,645 And that's what we'll see. 42 00:02:20,645 --> 00:02:24,138 You could do this in a number of ways that if you don't 43 00:02:24,138 --> 00:02:25,136 [INAUDIBLE] 44 00:02:25,136 --> 00:02:27,132 that gamma would turn out to be a 45 00:02:27,132 --> 00:02:28,629 crystallographic rotation. 46 00:02:28,629 --> 00:02:32,454 And then your result would be true, but it would not be a 47 00:02:32,454 --> 00:02:35,116 rotation operation which could exist in a three-dimensional 48 00:02:35,116 --> 00:02:36,613 point group. 49 00:02:36,613 --> 00:02:42,601 So using the genius of Leonhard Euler and a 50 00:02:42,601 --> 00:02:45,096 construction known as Euler's construction. 51 00:02:45,096 --> 00:02:49,088 We set up a little spherical triangle, 52 00:02:49,088 --> 00:02:52,581 which we could analyze. 53 00:02:52,581 --> 00:02:55,575 Let me tell you a little bit about Euler, because he's a 54 00:02:55,575 --> 00:02:57,571 remarkable individual. 55 00:02:57,571 --> 00:03:02,561 The first remarkable feature of Euler is that he's Swiss, 56 00:03:02,561 --> 00:03:06,303 and there are not many world-class famous people who 57 00:03:06,303 --> 00:03:09,547 are Swiss, simply because the population is so small. 58 00:03:09,547 --> 00:03:13,040 The probability of somebody rising to heights is constant 59 00:03:13,040 --> 00:03:14,537 among all populations. 60 00:03:14,537 --> 00:03:16,731 If you have a small population, there are not 61 00:03:16,731 --> 00:03:18,030 going to be very many. 62 00:03:18,030 --> 00:03:21,220 And to demonstrate that point, can somebody identify some 63 00:03:21,220 --> 00:03:24,984 other citizen of Switzerland who rose to great heights, as 64 00:03:24,984 --> 00:03:27,414 world-famous as [INAUDIBLE]? 65 00:03:27,414 --> 00:03:30,330 Think of one other person? 66 00:03:30,330 --> 00:03:32,250 I'm fairly pressed to do so myself. 67 00:03:32,250 --> 00:03:33,844 There's my uncle, but he actually 68 00:03:33,844 --> 00:03:36,214 didn't amount to much. 69 00:03:36,214 --> 00:03:41,278 But there's an artist, Paul Klee, who is world-class. 70 00:03:41,278 --> 00:03:44,986 He was one of the early modern artists. 71 00:03:44,986 --> 00:03:48,458 And Switzerland has just finished constructing a 72 00:03:48,458 --> 00:03:51,599 marvelous museum on the outskirts of the 73 00:03:51,599 --> 00:03:52,426 capital city, Berne. 74 00:03:52,426 --> 00:03:55,898 It's a structure that is supposed to mimic the rolling 75 00:03:55,898 --> 00:03:58,378 countryside of the central part of Switzerland. 76 00:03:58,378 --> 00:04:02,594 So it's a series of cylindrical structures, glass 77 00:04:02,594 --> 00:04:04,330 in front, glass on top. 78 00:04:04,330 --> 00:04:09,315 And it divides the area into three. 79 00:04:09,315 --> 00:04:12,952 Two of them are exhibit spaces and one is a space for 80 00:04:12,952 --> 00:04:14,165 scholars and researchers. 81 00:04:14,165 --> 00:04:17,075 And it's an absolutely marvelous structure. 82 00:04:17,075 --> 00:04:19,985 It appeared in the pages of Time Magazine when it opened 83 00:04:19,985 --> 00:04:22,410 [INAUDIBLE] about two months ago. 84 00:04:22,410 --> 00:04:26,622 Anyway, Euler was born in 1707, so he 85 00:04:26,622 --> 00:04:28,600 operated a long time ago. 86 00:04:28,600 --> 00:04:36,026 And he died in St. Petersburg on September 18, 1783. 87 00:04:36,026 --> 00:04:43,834 That is exactly 350 years and one day after my birthday. 88 00:04:46,762 --> 00:04:48,830 That's another remarkable thing about him. 89 00:04:48,830 --> 00:04:51,387 He was 76 years old when he died. 90 00:04:51,387 --> 00:04:55,711 And that time of primitive medicine and plague, not many 91 00:04:55,711 --> 00:05:00,868 people got to live to their 60s and 70s. 92 00:05:00,868 --> 00:05:07,355 Euler studied at the University of Basel under the 93 00:05:07,355 --> 00:05:07,854 Bernoullis. 94 00:05:07,854 --> 00:05:10,220 I think you've all heard of the Bernoullis. 95 00:05:10,220 --> 00:05:13,548 There's a very famous principle of physics known as 96 00:05:13,548 --> 00:05:16,910 the Bernoulli effect, which stated in its simple practical 97 00:05:16,910 --> 00:05:20,108 form says that if you have the Sunday paper on the front seat 98 00:05:20,108 --> 00:05:23,306 alongside of you, and you drive your car with the 99 00:05:23,306 --> 00:05:26,000 windows down, the paper will blow out the window. 100 00:05:26,000 --> 00:05:29,810 That's Bernoulli's principle in action. 101 00:05:29,810 --> 00:05:36,228 Euler got his doctorate from Basel at age 16. 102 00:05:36,228 --> 00:05:40,218 It sort of leads one to the rhetorical question, how come 103 00:05:40,218 --> 00:05:43,608 you guys have been spinning your wheels for so long? 104 00:05:43,608 --> 00:05:47,052 But then I said, he was an unusual individual. 105 00:05:47,052 --> 00:05:52,464 The Bernoullis went to St. Petersburg in Russia, under 106 00:05:52,464 --> 00:05:54,924 Catherine the First. 107 00:05:54,924 --> 00:05:58,860 Russia was trying very hard at that time to enter the ranks 108 00:05:58,860 --> 00:06:02,340 of the Western world as a full member. 109 00:06:02,340 --> 00:06:04,630 Euler followed them a little bit later on. 110 00:06:04,630 --> 00:06:08,180 And he succeeded one of the Bernoullis as a professor of 111 00:06:08,180 --> 00:06:11,568 mathematics in 1733. 112 00:06:11,568 --> 00:06:16,970 Then unfortunately, two years later in 1735, he lost the 113 00:06:16,970 --> 00:06:18,100 sight of one eye. 114 00:06:18,100 --> 00:06:19,071 And why? 115 00:06:19,071 --> 00:06:22,017 Because at that time, astronomy had been using this 116 00:06:22,017 --> 00:06:25,945 newfangled telescope which had recently been perfected. 117 00:06:25,945 --> 00:06:28,400 One of the hottest things going was studying the heavens 118 00:06:28,400 --> 00:06:30,364 looking through a telescope. 119 00:06:30,364 --> 00:06:33,358 And if you wanted to look at the sun, people knew nothing 120 00:06:33,358 --> 00:06:36,680 about the damaging effect on retinas of the sun's rays. 121 00:06:36,680 --> 00:06:39,810 So he lost the sight of one eye. 122 00:06:39,810 --> 00:06:43,424 1741, he went back to Europe again, to Berlin. 123 00:06:43,424 --> 00:06:44,816 Why? 124 00:06:44,816 --> 00:06:48,435 Because the reigning monarch in Russia at that time was 125 00:06:48,435 --> 00:06:53,405 called Ivan the Terrible, and that says reams about why 126 00:06:53,405 --> 00:06:55,959 Euler would want to get out of Russia. 127 00:06:55,959 --> 00:06:59,871 But then 1776, he went back to St. Petersburg under the next 128 00:06:59,871 --> 00:07:01,827 monarch, who was Catherine the Great. 129 00:07:01,827 --> 00:07:04,272 And that says why one would be interested to go back. 130 00:07:04,272 --> 00:07:09,162 Finally, in 1766, he went fully blind. 131 00:07:09,162 --> 00:07:10,629 Did that slow him down? 132 00:07:10,629 --> 00:07:13,074 Not one bit. 133 00:07:13,074 --> 00:07:17,964 He published in his lifetime 800 papers. 134 00:07:17,964 --> 00:07:20,712 You talk to some big cheese around MIT, they've published 135 00:07:20,712 --> 00:07:26,965 maybe 200 or 300 papers, and that with the assistance of an 136 00:07:26,965 --> 00:07:30,918 army of graduate students, and also, one might add, the 137 00:07:30,918 --> 00:07:34,341 assistance of Xerox machines and word processors. 138 00:07:34,341 --> 00:07:37,275 So back in the days when you wrote everything out by hand 139 00:07:37,275 --> 00:07:41,187 with a [INAUDIBLE] quill pen, 800 papers is an absolutely 140 00:07:41,187 --> 00:07:44,121 unbelievable accomplishment. 141 00:07:44,121 --> 00:07:47,560 It took 35 years after he passed away to publish 142 00:07:47,560 --> 00:07:49,544 everything that he'd written. 143 00:07:49,544 --> 00:07:53,020 People had kept busy publishing what he did. 144 00:07:53,020 --> 00:07:56,812 And among his accomplishments, he was one of the first people 145 00:07:56,812 --> 00:08:00,703 to apply real hardcore mathematics to astronomy, to 146 00:08:00,703 --> 00:08:02,870 make it quantitative. 147 00:08:02,870 --> 00:08:06,975 He was one of the first to suggest that light was a wave 148 00:08:06,975 --> 00:08:10,602 form, and that color was a function of wave length. 149 00:08:10,602 --> 00:08:13,398 That was astonishingly precocious. 150 00:08:13,398 --> 00:08:16,917 And then, lest he seem like an egghead who spent all his time 151 00:08:16,917 --> 00:08:21,318 staring through telescopes and working out theorems to use in 152 00:08:21,318 --> 00:08:24,741 crystallography, he also wrote a popular account of science 153 00:08:24,741 --> 00:08:28,164 for the general public, which was published in 1768. 154 00:08:28,164 --> 00:08:32,565 And that book was published for 90 years, three 155 00:08:32,565 --> 00:08:35,499 generations of people kept gobbling up [INAUDIBLE] 156 00:08:35,499 --> 00:08:37,000 pretty good. 157 00:08:37,000 --> 00:08:40,980 He impinged upon our own language and activities in 158 00:08:40,980 --> 00:08:42,030 several important ways. 159 00:08:42,030 --> 00:08:47,170 He was the one who used lowercase i to define the 160 00:08:47,170 --> 00:08:48,060 square root of minus 1. 161 00:08:48,060 --> 00:08:51,140 We can thank Euler for that. 162 00:08:51,140 --> 00:08:55,948 He was the person who used e to define the constant, 163 00:08:55,948 --> 00:09:06,406 2.71828182845904523536. 164 00:09:06,406 --> 00:09:09,394 And he was the person who first used f 165 00:09:09,394 --> 00:09:11,386 to stand for function. 166 00:09:11,386 --> 00:09:15,370 So he contributed not only a lot of good mathematics, but a 167 00:09:15,370 --> 00:09:17,362 lot [INAUDIBLE]. 168 00:09:17,362 --> 00:09:19,720 So this does not have to be easy. 169 00:09:19,720 --> 00:09:20,905 Euler was a great guy. 170 00:09:20,905 --> 00:09:27,720 And this geometry of rotations about different axes is 171 00:09:27,720 --> 00:09:31,712 something that also survives in a mechanism that involves 172 00:09:31,712 --> 00:09:37,495 achieving angular core rotation on a axis by rotation 173 00:09:37,495 --> 00:09:39,836 on two orthogonal arcs. 174 00:09:39,836 --> 00:09:42,800 And that's something that's called an Euler Cradle. 175 00:09:47,246 --> 00:09:51,251 And that is geometry that;s used in a great number of 176 00:09:51,251 --> 00:09:52,652 mechanical devices. 177 00:09:52,652 --> 00:09:56,610 In any case, back to instruction for our purposes. 178 00:09:56,610 --> 00:10:02,126 The thing that we would like to do is let alpha and beta 179 00:10:02,126 --> 00:10:05,770 take on all possible crystallographic values, 180 00:10:05,770 --> 00:10:10,711 namely 360 degrees or onefold axis, although we know that 181 00:10:10,711 --> 00:10:12,458 that's not gonna work. 182 00:10:12,458 --> 00:10:14,454 Twofold, 180 degrees. 183 00:10:14,454 --> 00:10:15,951 Threefold, 120. 184 00:10:15,951 --> 00:10:16,949 Fourfold, 90. 185 00:10:16,949 --> 00:10:20,442 Sixfold, 60. 186 00:10:20,442 --> 00:10:24,933 And let that give the values to alpha and beta, 187 00:10:24,933 --> 00:10:26,430 taking two at a time. 188 00:10:26,430 --> 00:10:30,422 And then let us ask the question, at what angle should 189 00:10:30,422 --> 00:10:35,412 we combine these two axes to get gamma to be a 190 00:10:35,412 --> 00:10:38,406 crystallographic rotation axis? 191 00:10:38,406 --> 00:10:44,394 And if it is crystallographic and not something like 37.9234 192 00:10:44,394 --> 00:10:47,887 degrees, what are the remaining axes with interaxial 193 00:10:47,887 --> 00:10:50,881 angles B and A? 194 00:10:50,881 --> 00:10:53,875 So this is the problem that Euler's construction solved. 195 00:10:53,875 --> 00:10:56,869 And I won't go through all the arguments that we 196 00:10:56,869 --> 00:10:57,867 need to set this up. 197 00:10:57,867 --> 00:11:02,857 But what we found after some f sleight of hand when we were 198 00:11:02,857 --> 00:11:06,350 working on the polar triangle with spherical trigonometry, 199 00:11:06,350 --> 00:11:11,340 what we found was the result that said that if we want to 200 00:11:11,340 --> 00:11:15,082 combine two axes, alpha and beta, so that the third one 201 00:11:15,082 --> 00:11:19,110 turned out to be a rotation of gamma, then the cosine of the 202 00:11:19,110 --> 00:11:24,130 angle between A and B should be the cosine of alpha/2, 203 00:11:24,130 --> 00:11:30,609 cosine of beta/2 plus cosine of gamma/2, divided by the 204 00:11:30,609 --> 00:11:35,900 sine of alpha/2, sine beta/2. 205 00:11:35,900 --> 00:11:41,259 So if you pick your alpha and beta, and you decide what you 206 00:11:41,259 --> 00:11:45,251 would want these first two rotations to turn out to be. 207 00:11:45,251 --> 00:11:46,748 And generally it's not gonna work. 208 00:11:46,748 --> 00:11:48,744 But there are a surprising number of cases 209 00:11:48,744 --> 00:11:50,241 where it does work. 210 00:11:50,241 --> 00:11:53,235 So you specify the combination you had. 211 00:11:53,235 --> 00:11:58,058 You also determine the angle between A and C. And you have 212 00:11:58,058 --> 00:12:02,716 to also determine the angle between the axes B and C. And 213 00:12:02,716 --> 00:12:06,209 there are similar sorts of expressions that one obtains 214 00:12:06,209 --> 00:12:11,199 simply by [INAUDIBLE] alpha and beta again. 215 00:12:11,199 --> 00:12:15,191 So then we set it up just by looking at all possible 216 00:12:15,191 --> 00:12:19,682 combinations of twofold, of two different rotation axes, 217 00:12:19,682 --> 00:12:23,190 and a third, which the net effect might be. 218 00:12:23,190 --> 00:12:26,836 We're not interested in permuting A, B, and C. And A 219 00:12:26,836 --> 00:12:29,752 equal to C equal to B is just as interesting or not as A 220 00:12:29,752 --> 00:12:33,640 equal to B equal to C. We don't care about permutations. 221 00:12:33,640 --> 00:12:38,500 And we generated-- just as we [INAUDIBLE] 222 00:12:38,500 --> 00:12:43,386 a week and a half ago-- a set of combinations that we should 223 00:12:43,386 --> 00:12:49,870 consider, what the axis A would be, what the axis B 224 00:12:49,870 --> 00:12:54,086 would be, and them different choices for the axis C. So A 225 00:12:54,086 --> 00:12:54,582 could be 1. 226 00:12:54,582 --> 00:12:56,070 B could be 1. 227 00:12:56,070 --> 00:13:00,534 And we could look for 1, 1, 1; 1, 1, 2; 1, 1, 3; 228 00:13:00,534 --> 00:13:03,510 1, 1, 4; 1, 1, 6. 229 00:13:03,510 --> 00:13:05,990 Those are legitimate combinations? 230 00:13:05,990 --> 00:13:10,206 Those are absurd combinations, because doing nothing about 231 00:13:10,206 --> 00:13:12,934 the first onefold axis, doing nothing about the second 232 00:13:12,934 --> 00:13:17,646 onefold axis could hardly result in the net effect of 233 00:13:17,646 --> 00:13:21,366 the 90-degree rotation by the third axis. 234 00:13:21,366 --> 00:13:24,838 And [INAUDIBLE] suggested is that sitting around and doing 235 00:13:24,838 --> 00:13:27,814 nothing twice was equal to a rotation [INAUDIBLE] its 236 00:13:27,814 --> 00:13:30,294 junctures [INAUDIBLE] we'd find ourselves spinning on our 237 00:13:30,294 --> 00:13:31,544 axis like tops. 238 00:13:36,742 --> 00:13:37,734 Twofold axis. 239 00:13:37,734 --> 00:13:40,462 1, 1, 2, we have here, so we don't have to 240 00:13:40,462 --> 00:13:42,198 consider 2, 1, 1. 241 00:13:42,198 --> 00:13:50,254 But we should consider 2, 1, 2; 2, 1, 3; 2, 1, 4; 2, 242 00:13:50,254 --> 00:13:53,206 1, 6, and so on. 243 00:13:53,206 --> 00:13:56,035 If we filled out this whole table, last time you got a 244 00:13:56,035 --> 00:14:00,832 copy of it and some notes, and all that remains then is to 245 00:14:00,832 --> 00:14:02,080 quote [INAUDIBLE] 246 00:14:02,080 --> 00:14:07,570 and see where we get allowable axial combinations. 247 00:14:07,570 --> 00:14:14,005 And not surprisingly, there are so very, very few. 248 00:14:14,005 --> 00:14:17,965 And we showed-- again, when we finished up last time-- that 249 00:14:17,965 --> 00:14:24,320 you can always take any n-fold axis that has a C gamma that's 250 00:14:24,320 --> 00:14:31,378 equal to C 2 pi over n and combine it with twofold axes 251 00:14:31,378 --> 00:14:36,834 at right angles, provided you make the angle between the 252 00:14:36,834 --> 00:14:42,786 twofold axes equal 2 of gamma over 2. 253 00:14:42,786 --> 00:14:47,498 So the crystallographic possibilities for C are, first 254 00:14:47,498 --> 00:14:52,706 of all, C could be a twofold axis, in which case you could 255 00:14:52,706 --> 00:14:57,170 combine with it a pair of twofold axes, and this 1/2 of 256 00:14:57,170 --> 00:15:00,146 180 degrees would also be a right angle. 257 00:15:04,610 --> 00:15:11,058 We could let C be a threefold axis, in which case the 258 00:15:11,058 --> 00:15:14,034 twofold axes have to be at right angles, two- to 259 00:15:14,034 --> 00:15:15,522 threefold axes. 260 00:15:15,522 --> 00:15:19,490 And they should be combined at half the angular throw of the 261 00:15:19,490 --> 00:15:23,458 threefold axes, which is 60 degrees. 262 00:15:23,458 --> 00:15:28,543 Two more possibilities are four [INAUDIBLE] pair of 263 00:15:28,543 --> 00:15:30,539 twofold axes at right angles. 264 00:15:30,539 --> 00:15:33,865 Of the angle between them, half of the throw of a 265 00:15:33,865 --> 00:15:36,527 fourfold axis would have to be equal to 5 degrees. 266 00:15:36,527 --> 00:15:40,519 And the last one is sixfold axis with a 267 00:15:40,519 --> 00:15:42,016 pair of twofold axes. 268 00:15:42,016 --> 00:15:43,513 Add angles to it. 269 00:15:43,513 --> 00:15:46,507 And a third [INAUDIBLE]. 270 00:15:46,507 --> 00:15:51,996 So notice the insidious fact that the angle between the 271 00:15:51,996 --> 00:15:58,982 twofold axes is always a half, 1/2 the rotation angle in 272 00:15:58,982 --> 00:16:04,471 principal axis C are not equal to this rotation axis. 273 00:16:04,471 --> 00:16:10,459 The other thing we saw that is that these twofold axes are 274 00:16:10,459 --> 00:16:15,845 different, distinct, symmetry-independent axes. 275 00:16:15,845 --> 00:16:20,795 They're different in that the principal axis C never rotates 276 00:16:20,795 --> 00:16:24,095 this axis into the second one, and therefore demands that 277 00:16:24,095 --> 00:16:28,343 whatever's going on around one twofold axis be identical to 278 00:16:28,343 --> 00:16:32,180 what's going on at the other twofold axis, different in the 279 00:16:32,180 --> 00:16:36,635 sense that they function in different ways in the pattern, 280 00:16:36,635 --> 00:16:41,090 or if you're describing the symmetry of an object. 281 00:16:41,090 --> 00:16:44,948 So probably the best demonstration of this is a 282 00:16:44,948 --> 00:16:50,936 regular prism with a triangular shape or with a 283 00:16:50,936 --> 00:16:56,924 square shape or with a hexagonal shape. 284 00:16:59,918 --> 00:17:04,409 And the adjacent twofold axes for these prisms would come 285 00:17:04,409 --> 00:17:09,399 out of the normal to a face. 286 00:17:09,399 --> 00:17:15,886 And then if the second twofold axis is going to be 45 degrees 287 00:17:15,886 --> 00:17:22,872 away from the first, the other one has to come [INAUDIBLE]. 288 00:17:22,872 --> 00:17:25,367 So yeah, they function in different ways in the space. 289 00:17:25,367 --> 00:17:28,194 One is normal to the face of a regular prism, if that's 290 00:17:28,194 --> 00:17:29,359 what's in our space. 291 00:17:29,359 --> 00:17:31,355 The other one comes out of the edge. 292 00:17:31,355 --> 00:17:40,087 Similarly for a sixfold axis, a hexagonal prism has one 293 00:17:40,087 --> 00:17:43,331 twofold axis coming out normal to a face, the adjacent 294 00:17:43,331 --> 00:17:45,327 twofold axis coming out to an edge. 295 00:17:45,327 --> 00:17:48,321 And as advertised, that angle is 33. 296 00:17:48,321 --> 00:17:50,317 So they function in different ways. 297 00:17:50,317 --> 00:17:53,250 The only exception to that, again, is the [INAUDIBLE] 298 00:17:53,250 --> 00:17:54,690 threefold axis. 299 00:17:54,690 --> 00:17:59,010 And the twofold axes there, which were 60 degrees, come 300 00:17:59,010 --> 00:18:04,706 out of one side from a corner of the triangular prism. 301 00:18:04,706 --> 00:18:06,698 On the other side, that was [INAUDIBLE]. 302 00:18:06,698 --> 00:18:10,682 So all of the twofold axes were the same thing. 303 00:18:10,682 --> 00:18:15,164 There's only one independent kind of twofold axis, just as 304 00:18:15,164 --> 00:18:18,650 there was only kind of mirror plane in the combination of a 305 00:18:18,650 --> 00:18:21,638 mirror plane passing through a twofold axis. 306 00:18:21,638 --> 00:18:25,855 The names for these are always, as we've done in the 307 00:18:25,855 --> 00:18:29,348 past, a running list of the independent symmetry 308 00:18:29,348 --> 00:18:30,845 operations that are present. 309 00:18:30,845 --> 00:18:36,334 So this general one, n, 2, 2, with the n-fold axis for some 310 00:18:36,334 --> 00:18:38,330 generic sort of a twofold axis. 311 00:18:38,330 --> 00:18:40,825 This one would be 2, 2, 2. 312 00:18:40,825 --> 00:18:43,320 This one would be 4, 2, 2. 313 00:18:43,320 --> 00:18:46,813 And this one would be 6, 2, 2. 314 00:18:46,813 --> 00:18:50,306 This [INAUDIBLE] with a threefold axis is, again, 315 00:18:50,306 --> 00:18:55,046 called not 3, 2, but 3, 2, because there's only one kind 316 00:18:55,046 --> 00:18:57,666 of twofold axis, just as there's only one kind 317 00:18:57,666 --> 00:18:58,916 [INAUDIBLE]. 318 00:19:03,779 --> 00:19:07,771 We pause [INAUDIBLE], see if you have any questions. 319 00:19:07,771 --> 00:19:10,266 These are the crystallographic combinations of this 320 00:19:10,266 --> 00:19:11,763 [INAUDIBLE]. 321 00:19:11,763 --> 00:19:17,252 There is no reason why you should not in something that 322 00:19:17,252 --> 00:19:20,495 doesn't have to be compatible with a lattice, combine an 323 00:19:20,495 --> 00:19:26,234 n-fold axis of any sort with twofold axes or right angles. 324 00:19:26,234 --> 00:19:29,727 And indeed, if I look at my old friend, the saguaro 325 00:19:29,727 --> 00:19:34,717 cactus, we can add anything like 28- 326 00:19:34,717 --> 00:19:38,210 up to 32-fold symmetry. 327 00:19:38,210 --> 00:19:42,202 This cactus stem had a 28-fold symmetry. 328 00:19:42,202 --> 00:19:46,318 It would be a twofold axis coming out of the string with 329 00:19:46,318 --> 00:19:50,186 one of the ribs, another twofold axis coming out of the 330 00:19:50,186 --> 00:19:52,681 crevice between the pair of these ribs. 331 00:19:52,681 --> 00:19:55,675 And if I took that thing up, very carefully because of the 332 00:19:55,675 --> 00:19:58,170 spines, and with great effort because it weights several 333 00:19:58,170 --> 00:20:04,657 tons, and rotated it about one axis, and then rotate it again 334 00:20:04,657 --> 00:20:07,152 about a second axis, turning it upside-down, [INAUDIBLE] 335 00:20:07,152 --> 00:20:11,144 axis would be rotation to 128 [INAUDIBLE]. 336 00:20:11,144 --> 00:20:13,639 Valid symmetry, but not crystallographic. 337 00:20:22,122 --> 00:20:26,120 OK, any comments or questions? 338 00:20:26,120 --> 00:20:29,708 Get to know these results, because the exercise that's 339 00:20:29,708 --> 00:20:35,518 going to occupy us for the next week is going to be 340 00:20:35,518 --> 00:20:39,004 asking how we can decorate these frameworks with mirror 341 00:20:39,004 --> 00:20:40,996 planes and with inversion. 342 00:20:40,996 --> 00:20:44,430 If you want orientation, we could add another operation to 343 00:20:44,430 --> 00:20:47,280 the collection of axes while it pops up. 344 00:20:47,280 --> 00:20:48,530 Where [INAUDIBLE]. 345 00:20:52,580 --> 00:20:53,830 Comments or questions? 346 00:20:58,030 --> 00:21:02,927 OK, there are only two other combinations that are 347 00:21:02,927 --> 00:21:07,330 crystallographic that involve directions 348 00:21:07,330 --> 00:21:08,736 that are not simple. 349 00:21:11,634 --> 00:21:23,060 And one of them involves a combination of a threefold 350 00:21:23,060 --> 00:21:29,534 axis with twofold axes that come out of directions at a 351 00:21:29,534 --> 00:21:31,028 normal to the face of a cube. 352 00:21:31,028 --> 00:21:34,763 So these turn out to be very, very strange angles which make 353 00:21:34,763 --> 00:21:38,000 no sense at all until you refer them to 354 00:21:38,000 --> 00:21:39,494 directions in the cube. 355 00:21:39,494 --> 00:21:41,984 The direction of the threefold axis turns out to be 356 00:21:41,984 --> 00:21:44,474 correspondent with the angle of a cube. 357 00:21:44,474 --> 00:21:47,628 The direction of the twofold axis corresponds to 358 00:21:47,628 --> 00:21:50,450 the normal two faces. 359 00:21:50,450 --> 00:21:51,446 You can show-- 360 00:21:51,446 --> 00:21:55,015 I did show, and I don't think anybody really followed me, so 361 00:21:55,015 --> 00:21:57,422 I had to hand out that [INAUDIBLE]-- 362 00:21:57,422 --> 00:22:01,406 that if you start with one twofold axis and one threefold 363 00:22:01,406 --> 00:22:05,639 axis, what you're going to get is a threefold axis coming out 364 00:22:05,639 --> 00:22:08,876 of all of the [INAUDIBLE] diagonals, but they're all 365 00:22:08,876 --> 00:22:12,860 equivalent to this threefold axis. 366 00:22:12,860 --> 00:22:16,844 And twofold axes come out to normal for all the 367 00:22:16,844 --> 00:22:17,840 faces of the cube. 368 00:22:17,840 --> 00:22:21,848 So there's only one kind of twofold axis and one kind of 369 00:22:21,848 --> 00:22:25,824 threefold axis, so even though we got this by combining a 370 00:22:25,824 --> 00:22:27,315 pair of twofold axes-- 371 00:22:27,315 --> 00:22:29,303 I'm sorry, a pair of threefold axes-- 372 00:22:32,782 --> 00:22:35,267 that's what happens when you stay up late. 373 00:22:35,267 --> 00:22:36,261 You forget about this one. 374 00:22:36,261 --> 00:22:37,752 A pair of threefold axes at the diagonals 375 00:22:37,752 --> 00:22:39,243 and one twofold axis. 376 00:22:39,243 --> 00:22:43,645 And this is the combination that is called 23, because 377 00:22:43,645 --> 00:22:47,328 there's only one sort of twofold axis and one sort of 378 00:22:47,328 --> 00:22:48,789 threefold axis. 379 00:22:48,789 --> 00:22:53,590 And I'd like to point out and I'd like to warn you of traps 380 00:22:53,590 --> 00:22:57,544 when we come across them, make sure we don't [INAUDIBLE] 381 00:22:57,544 --> 00:22:58,038 across them. 382 00:22:58,038 --> 00:23:04,460 Notice the insidious relation of this pair of integers to 383 00:23:04,460 --> 00:23:07,918 the symmetry that we label [? 232. ?] 384 00:23:07,918 --> 00:23:12,858 32 is a threefold axis with a twofold axis normal to it. 385 00:23:12,858 --> 00:23:16,316 23 is this combination that involves 386 00:23:16,316 --> 00:23:17,798 corrections in a cube. 387 00:23:22,738 --> 00:23:26,690 Now, let me pause parenthetically with an aside. 388 00:23:26,690 --> 00:23:29,160 You might say, how can this be? 389 00:23:29,160 --> 00:23:30,642 Here is a cube. 390 00:23:30,642 --> 00:23:35,100 That cube has got a fourfold axis about it. 391 00:23:35,100 --> 00:23:37,138 Don't call that a twofold axis. 392 00:23:37,138 --> 00:23:40,072 A cube has a fourfold axis coming out of it. 393 00:23:40,072 --> 00:23:43,495 OK, let me give you an example in real life. 394 00:23:43,495 --> 00:23:48,874 There is a fairly common mineral, iron disulfide-- 395 00:23:48,874 --> 00:23:50,840 pyrite. 396 00:23:50,840 --> 00:23:54,913 This forms nice, shiny cubes, but the cube faces have 397 00:23:54,913 --> 00:23:56,380 striations on them. 398 00:23:56,380 --> 00:23:59,314 If you look at them, there's a set of lines running this way. 399 00:24:02,750 --> 00:24:05,960 And what those lines are if you look at this crystal face 400 00:24:05,960 --> 00:24:10,672 with a magnifying glass, is that these are little steps of 401 00:24:10,672 --> 00:24:11,912 a second face. 402 00:24:11,912 --> 00:24:14,392 And this is a face of the form hk0. 403 00:24:14,392 --> 00:24:17,368 And this oscillates back and forth, and there seem to be 404 00:24:17,368 --> 00:24:19,848 lines scribed on the surface. 405 00:24:19,848 --> 00:24:22,840 So this sort of a [INAUDIBLE] 406 00:24:22,840 --> 00:24:25,362 crystal growth of one face which never really develops is 407 00:24:25,362 --> 00:24:27,060 not that uncommon. 408 00:24:27,060 --> 00:24:29,970 There's a threefold axis coming out of the corner here, 409 00:24:29,970 --> 00:24:34,577 but this is really a surface that is left at variant only 410 00:24:34,577 --> 00:24:36,760 by 183 [INAUDIBLE]. 411 00:24:36,760 --> 00:24:39,200 You cannot rotate that surface 90 degrees. 412 00:24:39,200 --> 00:24:41,438 The orientation of the lines have changed. 413 00:24:41,438 --> 00:24:44,681 But there is a bona fide threefold axis coming up 414 00:24:44,681 --> 00:24:50,919 there, so these striations, if I rotate them to this face, 415 00:24:50,919 --> 00:24:52,915 will go in a way like this. 416 00:24:52,915 --> 00:24:57,239 This edge turns into this edge, and therefore the lines 417 00:24:57,239 --> 00:24:58,404 will run down like this. 418 00:24:58,404 --> 00:25:01,897 And if I rotate [INAUDIBLE] n by 90 degrees, the striations 419 00:25:01,897 --> 00:25:07,885 on the adjacent face will run down. 420 00:25:07,885 --> 00:25:09,881 So there's a decorated cube. 421 00:25:09,881 --> 00:25:12,625 And if you rolled it up and say how can I move this cube 422 00:25:12,625 --> 00:25:15,869 around and leave its appearance totally unchanged? 423 00:25:15,869 --> 00:25:19,861 the answer is, rotate it by 120 degrees [INAUDIBLE] 424 00:25:19,861 --> 00:25:20,360 diagonal. 425 00:25:20,360 --> 00:25:24,352 But we can only rotate it 180 degrees around the face. 426 00:25:24,352 --> 00:25:27,346 So there's an example of a crystal [INAUDIBLE] 427 00:25:27,346 --> 00:25:30,839 on the arrangement of rotation axes 23. 428 00:25:36,328 --> 00:25:43,314 The final one, the highest symmetry of all, involves a 429 00:25:43,314 --> 00:25:50,300 fourfold axis coming out of a direction normal to a face, a 430 00:25:50,300 --> 00:25:53,793 threefold axis coming out of the [INAUDIBLE] diagonal, and 431 00:25:53,793 --> 00:25:57,286 a twofold axis coming out normal to an edge. 432 00:25:57,286 --> 00:26:01,111 And if you let these axes work on one another, there's a 433 00:26:01,111 --> 00:26:03,773 twofold axis that comes out of every edge, and a fourfold 434 00:26:03,773 --> 00:26:05,769 axis that comes out of every face. 435 00:26:05,769 --> 00:26:09,761 And this one is named 432, because it's a combination of 436 00:26:09,761 --> 00:26:12,256 a fourfold, a threefold, and a twofold. 437 00:26:12,256 --> 00:26:14,751 That is the symmetry to the cube. 438 00:26:14,751 --> 00:26:19,242 And for crystallographic symmetries, that's about as 439 00:26:19,242 --> 00:26:23,234 complicated as it gets. 440 00:26:23,234 --> 00:26:26,228 Now, if we look at the regular solids that we've encountered 441 00:26:26,228 --> 00:26:32,715 here, with symmetry 23, there is a regular [INAUDIBLE] 442 00:26:32,715 --> 00:26:34,711 consisting of four triangular faces. 443 00:26:34,711 --> 00:26:35,961 That's a tetrahedron. 444 00:26:41,740 --> 00:26:47,153 And for 432, one of the polyhedra that can form from 445 00:26:47,153 --> 00:26:51,327 the crystal [INAUDIBLE] is an octahedron. 446 00:26:51,327 --> 00:26:57,219 These were the lovely solids called Platonic solids, after 447 00:26:57,219 --> 00:27:02,129 Plato, that we used as our trophies early this afternoon. 448 00:27:02,129 --> 00:27:03,602 So this is an octahedron. 449 00:27:07,530 --> 00:27:11,900 Let me finish up before our break by asking is there any 450 00:27:11,900 --> 00:27:16,631 other regular polyhedra that can result from a combination 451 00:27:16,631 --> 00:27:19,370 of rotation axes that are not crystallographic? 452 00:27:24,350 --> 00:27:27,350 Now, that's a tough question to ask. 453 00:27:27,350 --> 00:27:33,360 You instantly scan your knowledge of geometry. 454 00:27:33,360 --> 00:27:36,240 Clearly, there are a lot of prisms [INAUDIBLE] 455 00:27:36,240 --> 00:27:38,160 infinite number of prisms [INAUDIBLE] n22. 456 00:27:38,160 --> 00:27:41,730 But there's only one other combination of axes 457 00:27:41,730 --> 00:27:44,688 non-crystallographic which results in a regular 458 00:27:44,688 --> 00:27:46,167 polyhedron. 459 00:27:46,167 --> 00:27:50,357 And this is a combination, believe it or not, of a 460 00:27:50,357 --> 00:27:54,548 fivefold axis with a threefold axis. 461 00:28:00,957 --> 00:28:04,901 This is a fivefold axis and a threefold axis 462 00:28:04,901 --> 00:28:08,850 and a twofold axis. 463 00:28:08,850 --> 00:28:13,060 And these result in a regular solid called an icosahedron. 464 00:28:20,670 --> 00:28:22,334 And that is so complicated that I won't 465 00:28:22,334 --> 00:28:24,310 attempt to draw it. 466 00:28:24,310 --> 00:28:26,780 But having said so, it looks like this. 467 00:28:26,780 --> 00:28:29,250 It has diamond-shaped faces. 468 00:28:29,250 --> 00:28:34,200 And there are five of these that come together in a 469 00:28:34,200 --> 00:28:35,450 [INAUDIBLE] form. 470 00:28:38,136 --> 00:28:40,596 So here's one of the [INAUDIBLE] diamond-shaped 471 00:28:40,596 --> 00:28:43,712 faces, another diamond-shaped face, and then the two other 472 00:28:43,712 --> 00:28:45,516 ones that come in like this. 473 00:28:45,516 --> 00:28:49,452 So there are 1, 2, 3, 4, 5 faces, so this is the 474 00:28:49,452 --> 00:28:51,420 orientation of a fivefold axis. 475 00:28:51,420 --> 00:28:55,356 The twofold axis comes out of a place where two of these 476 00:28:55,356 --> 00:28:57,324 edges are shared. 477 00:28:57,324 --> 00:29:00,840 And the threefold axis-- 478 00:29:00,840 --> 00:29:02,313 [INAUDIBLE] triangular faces. 479 00:29:05,259 --> 00:29:06,509 [INAUDIBLE]. 480 00:29:10,660 --> 00:29:12,624 So here's the fivefold axis. 481 00:29:12,624 --> 00:29:14,588 These are the twofold axes [INAUDIBLE]. 482 00:29:14,588 --> 00:29:16,552 And these are threefold axes. 483 00:29:16,552 --> 00:29:18,730 But you know all this. 484 00:29:18,730 --> 00:29:21,307 Is there anybody who [INAUDIBLE] 485 00:29:21,307 --> 00:29:25,798 show me that they've never seen an icosahedron? 486 00:29:25,798 --> 00:29:28,293 Anybody ever who has not seen-- have you seen it? 487 00:29:28,293 --> 00:29:29,543 [INAUDIBLE]. 488 00:29:32,784 --> 00:29:33,283 Here-- 489 00:29:33,283 --> 00:29:37,024 I spared no expense-- is a live icosahedron for those who 490 00:29:37,024 --> 00:29:38,274 would like to look. 491 00:29:44,760 --> 00:29:48,752 AUDIENCE: How many sides does it have? 492 00:29:48,752 --> 00:29:50,002 PROFESSOR: He'll count them for you. 493 00:29:52,744 --> 00:29:53,243 I don't remember. 494 00:29:53,243 --> 00:29:55,488 I know the number of faces and the number of edges, but not 495 00:29:55,488 --> 00:29:58,732 the [INAUDIBLE]. 496 00:29:58,732 --> 00:30:00,229 AUDIENCE: [INAUDIBLE] 497 00:30:00,229 --> 00:30:00,728 faces. 498 00:30:00,728 --> 00:30:02,724 PROFESSOR: I think it's-- 499 00:30:02,724 --> 00:30:04,221 I'm not sure [INAUDIBLE]. 500 00:30:04,221 --> 00:30:05,219 Not sure. 501 00:30:05,219 --> 00:30:06,217 AUDIENCE: But you said you know the number of faces. 502 00:30:06,217 --> 00:30:07,714 PROFESSOR: Yeah, but I can't tell you everything I know 503 00:30:07,714 --> 00:30:08,213 [INAUDIBLE]. 504 00:30:08,213 --> 00:30:10,209 AUDIENCE: Yeah, yeah. 505 00:30:10,209 --> 00:30:14,201 PROFESSOR: There's another figure which also has this 506 00:30:14,201 --> 00:30:16,696 symmetry, and it's a regular solid. 507 00:30:16,696 --> 00:30:19,191 And this is called a rhombic dodecahedron. 508 00:30:27,770 --> 00:30:30,468 And, wise guy, this has 12 faces. 509 00:30:30,468 --> 00:30:33,444 And the faces are pentagonal. 510 00:30:33,444 --> 00:30:39,148 And there are three pentagonal faces that come together at 511 00:30:39,148 --> 00:30:43,364 the threefold axis, a fivefold axis out of each of the 512 00:30:43,364 --> 00:30:48,820 pentagonal faces, and a twofold axis out of the edges. 513 00:30:48,820 --> 00:30:54,370 The face that this has 12 faces at regular intervals 514 00:30:54,370 --> 00:30:58,562 leads an entrepreneur who was familiar with injection 515 00:30:58,562 --> 00:31:03,030 molding to cast these little things as a plastic, and then 516 00:31:03,030 --> 00:31:07,169 puts a month of the year on each of the 12 faces and made 517 00:31:07,169 --> 00:31:09,930 a nice little desk calendar. 518 00:31:09,930 --> 00:31:12,888 [INAUDIBLE] something that reminds you of symmetry as 519 00:31:12,888 --> 00:31:14,860 well [INAUDIBLE]. 520 00:31:14,860 --> 00:31:23,838 So this has fivefold faces, 12 of them, and that's the 521 00:31:23,838 --> 00:31:26,328 rhombic dodecahedron as opposed to the normal 522 00:31:26,328 --> 00:31:28,320 dodecahedron which is [INAUDIBLE]. 523 00:31:33,798 --> 00:31:36,288 AUDIENCE: So are those both [INAUDIBLE]? 524 00:31:36,288 --> 00:31:37,284 PROFESSOR: I'm sorry? 525 00:31:37,284 --> 00:31:40,272 AUDIENCE: Are those both [INAUDIBLE]? 526 00:31:40,272 --> 00:31:42,264 The icosahedron, [INAUDIBLE]? 527 00:31:42,264 --> 00:31:45,501 PROFESSOR: Yeah, this has a fivefold axis coming out of 528 00:31:45,501 --> 00:31:47,244 the pentagonal [INAUDIBLE]. 529 00:31:47,244 --> 00:31:48,240 Is that what you're asking? 530 00:31:48,240 --> 00:31:51,477 And the threefold axis, there are three of them that, come 531 00:31:51,477 --> 00:31:55,212 together, and they do something like this. 532 00:32:00,192 --> 00:32:04,425 So here's the threefold, here's the fivefold, here's 533 00:32:04,425 --> 00:32:05,675 the twofold. 534 00:32:09,156 --> 00:32:11,148 Here we have to see it [INAUDIBLE]. 535 00:32:11,148 --> 00:32:14,634 Look for somebody who's got one of these desk calendars. 536 00:32:14,634 --> 00:32:16,128 [INAUDIBLE] used to sell them. 537 00:32:16,128 --> 00:32:17,378 [INAUDIBLE]. 538 00:32:23,140 --> 00:32:26,822 OK, this sets up the next stage of our game. 539 00:32:26,822 --> 00:32:29,873 We've got these arrangements of axes. 540 00:32:29,873 --> 00:32:34,803 And if you count them up on the fingers of your hands and 541 00:32:34,803 --> 00:32:37,761 one toe, there are 11 of them. 542 00:32:37,761 --> 00:32:41,705 There are the axes by themselves-- onefold, twofold, 543 00:32:41,705 --> 00:32:43,677 threefold, fourfold, sixfold. 544 00:32:43,677 --> 00:32:47,621 There are these so-called dihedral combinations, where 545 00:32:47,621 --> 00:32:51,565 the only thing that changes from one to the other is the 546 00:32:51,565 --> 00:32:55,509 symmetry of the main axis, the [INAUDIBLE] symmetry, and 547 00:32:55,509 --> 00:32:58,960 therefore the dihedral angle between the twofold axes. 548 00:32:58,960 --> 00:33:05,658 And these are 222, 32, 422, and 622. 549 00:33:05,658 --> 00:33:10,598 And then the two cubic arrangements, 32 and 432. 550 00:33:10,598 --> 00:33:16,190 So when we return, we'll ask the question, how can we add 551 00:33:16,190 --> 00:33:19,097 mirror planes for an inversion center to this 552 00:33:19,097 --> 00:33:20,350 combination of axes? 553 00:33:20,350 --> 00:33:23,367 And these are going to give us new symmetries involving not 554 00:33:23,367 --> 00:33:27,199 only rotation, but [INAUDIBLE] help with rotation 555 00:33:27,199 --> 00:33:30,073 inversion as well. 556 00:33:30,073 --> 00:33:35,226 And the constraint in doing this is that we have to add 557 00:33:35,226 --> 00:33:39,660 the reflection plane and the inversion center in such a way 558 00:33:39,660 --> 00:33:46,880 that it doesn't create any new rotation axes, because we have 559 00:33:46,880 --> 00:33:50,210 systematically derived all of the possible combinations of 560 00:33:50,210 --> 00:33:52,202 crystallographic rotation axes. 561 00:33:52,202 --> 00:33:56,186 So if the addition of a mirror plane, creates a new axis, 562 00:33:56,186 --> 00:34:00,668 that's going to be something that you already have with a 563 00:34:00,668 --> 00:34:03,158 combination of a greater number of rotation axes. 564 00:34:03,158 --> 00:34:05,648 For example, if you take a single twofold axis and a 565 00:34:05,648 --> 00:34:09,632 mirror plane of an angle, that generated another twofold axis 566 00:34:09,632 --> 00:34:11,126 90 degrees away. 567 00:34:11,126 --> 00:34:12,620 But we've already got that. 568 00:34:12,620 --> 00:34:19,094 If a mirror plane moves an axis to an angle that doesn't 569 00:34:19,094 --> 00:34:22,082 correspond to one of these arrangements, it's going to be 570 00:34:22,082 --> 00:34:25,568 impossible, because we have systematically derived, using 571 00:34:25,568 --> 00:34:29,054 Euler's construction, all the combinations of rotation 572 00:34:29,054 --> 00:34:31,046 operations that are possible. 573 00:34:31,046 --> 00:34:33,038 So that's going to be the constraint. 574 00:34:33,038 --> 00:34:37,520 We want to add mirror planes or an inversion center in all 575 00:34:37,520 --> 00:34:39,014 possible combinations. 576 00:34:39,014 --> 00:34:42,002 And this means we're gonna need a theorem. 577 00:34:42,002 --> 00:34:44,492 What happens when you add a mirror plane 578 00:34:44,492 --> 00:34:46,500 to a rotation operation? 579 00:34:46,500 --> 00:34:47,566 We're already familiar with one of them. 580 00:34:47,566 --> 00:34:52,030 You take an axis and you pass an mirror plane through it, 581 00:34:52,030 --> 00:34:55,502 you get another mirror plane that is rotated about the 582 00:34:55,502 --> 00:34:58,974 axis, away from the first, by half the rotation 583 00:34:58,974 --> 00:35:01,970 angle of the axis. 584 00:35:01,970 --> 00:35:06,520 So let's take a breather, and let us 585 00:35:06,520 --> 00:35:08,620 resume in about 10 minutes.