Recorded at | July 24, 2009 |
---|---|

Event | TEDGlobal 2009 |

Duration (min:sec) | 17:58 |

Video Type | TED Stage Talk |

Words per minute | 227.64 very fast |

Readability (FK) | 61.13 easy |

Speaker | Marcus du Sautoy |

Country | United Kingdom |

Occupation | mathematician, academic, writer |

Description | British professor of mathematics |

Official TED page for this talk

**Synopsis**

The world turns on symmetry -- from the spin of subatomic particles to the dizzying beauty of an arabesque. But there's more to it than meets the eye. Here, Oxford mathematician Marcus du Sautoy offers a glimpse of the invisible numbers that marry all symmetrical objects.

1 | 00:18 | On the 30th of May, 1832, a gunshot was heard ringing out across the 13th arrondissement in Paris. | ||

2 | 00:27 | (Gunshot) A peasant, who was walking to market that morning, ran towards where the gunshot had come from, and found a young man writhing in agony on the floor, clearly shot by a dueling wound. | ||

3 | 00:40 | The young man's name was Evariste Galois. | ||

4 | 00:43 | He was a well-known revolutionary in Paris at the time. | ||

5 | 00:47 | Galois was taken to the local hospital where he died the next day in the arms of his brother. | ||

6 | 00:53 | And the last words he said to his brother were, "Don't cry for me, Alfred. | ||

7 | 00:57 | I need all the courage I can muster to die at the age of 20." | ||

8 | 01:03 | It wasn't, in fact, revolutionary politics for which Galois was famous. | ||

9 | 01:07 | But a few years earlier, while still at school, he'd actually cracked one of the big mathematical problems at the time. | ||

10 | 01:14 | And he wrote to the academicians in Paris, trying to explain his theory. | ||

11 | 01:18 | But the academicians couldn't understand anything that he wrote. | ||

12 | 01:21 | (Laughter) | ||

13 | 01:22 | This is how he wrote most of his mathematics. | ||

14 | 01:25 | So, the night before that duel, he realized this possibly is his last chance to try and explain his great breakthrough. | ||

15 | 01:32 | So he stayed up the whole night, writing away, trying to explain his ideas. | ||

16 | 01:37 | And as the dawn came up and he went to meet his destiny, he left this pile of papers on the table for the next generation. | ||

17 | 01:44 | Maybe the fact that he stayed up all night doing mathematics was the fact that he was such a bad shot that morning and got killed. | ||

18 | 01:50 | But contained inside those documents was a new language, a language to understand one of the most fundamental concepts of science -- namely symmetry. | ||

19 | 02:00 | Now, symmetry is almost nature's language. | ||

20 | 02:02 | It helps us to understand so many different bits of the scientific world. | ||

21 | 02:06 | For example, molecular structure. | ||

22 | 02:08 | What crystals are possible, we can understand through the mathematics of symmetry. | ||

23 | 02:14 | In microbiology you really don't want to get a symmetrical object, because they are generally rather nasty. | ||

24 | 02:18 | The swine flu virus, at the moment, is a symmetrical object. | ||

25 | 02:21 | And it uses the efficiency of symmetry to be able to propagate itself so well. | ||

26 | 02:27 | But on a larger scale of biology, actually symmetry is very important, because it actually communicates genetic information. | ||

27 | 02:32 | I've taken two pictures here and I've made them artificially symmetrical. | ||

28 | 02:36 | And if I ask you which of these you find more beautiful, you're probably drawn to the lower two. | ||

29 | 02:41 | Because it is hard to make symmetry. | ||

30 | 02:44 | And if you can make yourself symmetrical, you're sending out a sign that you've got good genes, you've got a good upbringing and therefore you'll make a good mate. | ||

31 | 02:51 | So symmetry is a language which can help to communicate genetic information. | ||

32 | 02:56 | Symmetry can also help us to explain what's happening in the Large Hadron Collider in CERN. | ||

33 | 03:01 | Or what's not happening in the Large Hadron Collider in CERN. | ||

34 | 03:04 | To be able to make predictions about the fundamental particles we might see there, it seems that they are all facets of some strange symmetrical shape in a higher dimensional space. | ||

35 | 03:14 | And I think Galileo summed up, very nicely, the power of mathematics to understand the scientific world around us. | ||

36 | 03:20 | He wrote, "The universe cannot be read until we have learnt the language and become familiar with the characters in which it is written. | ||

37 | 03:27 | It is written in mathematical language, and the letters are triangles, circles and other geometric figures, without which means it is humanly impossible to comprehend a single word." | ||

38 | 03:38 | But it's not just scientists who are interested in symmetry. | ||

39 | 03:41 | Artists too love to play around with symmetry. | ||

40 | 03:44 | They also have a slightly more ambiguous relationship with it. | ||

41 | 03:47 | Here is Thomas Mann talking about symmetry in "The Magic Mountain." | ||

42 | 03:50 | He has a character describing the snowflake, and he says he "shuddered at its perfect precision, found it deathly, the very marrow of death." | ||

43 | 03:59 | But what artists like to do is to set up expectations of symmetry and then break them. | ||

44 | 04:03 | And a beautiful example of this I found, actually, when I visited a colleague of mine in Japan, Professor Kurokawa. | ||

45 | 04:09 | And he took me up to the temples in Nikko. | ||

46 | 04:12 | And just after this photo was taken we walked up the stairs. | ||

47 | 04:15 | And the gateway you see behind has eight columns, with beautiful symmetrical designs on them. | ||

48 | 04:20 | Seven of them are exactly the same, and the eighth one is turned upside down. | ||

49 | 04:25 | And I said to Professor Kurokawa, "Wow, the architects must have really been kicking themselves when they realized that they'd made a mistake and put this one upside down." | ||

50 | 04:32 | And he said, "No, no, no. It was a very deliberate act." | ||

51 | 04:35 | And he referred me to this lovely quote from the Japanese "Essays in Idleness" from the 14th century, in which the essayist wrote, "In everything, uniformity is undesirable. | ||

52 | 04:45 | Leaving something incomplete makes it interesting, and gives one the feeling that there is room for growth." | ||

53 | 04:50 | Even when building the Imperial Palace, they always leave one place unfinished. | ||

54 | 04:56 | But if I had to choose one building in the world to be cast out on a desert island, to live the rest of my life, being an addict of symmetry, I would probably choose the Alhambra in Granada. | ||

55 | 05:06 | This is a palace celebrating symmetry. | ||

56 | 05:08 | Recently I took my family -- we do these rather kind of nerdy mathematical trips, which my family love. | ||

57 | 05:13 | This is my son Tamer. You can see he's really enjoying our mathematical trip to the Alhambra. | ||

58 | 05:18 | But I wanted to try and enrich him. | ||

59 | 05:21 | I think one of the problems about school mathematics is it doesn't look at how mathematics is embedded in the world we live in. | ||

60 | 05:27 | So, I wanted to open his eyes up to how much symmetry is running through the Alhambra. | ||

61 | 05:32 | You see it already. Immediately you go in, the reflective symmetry in the water. | ||

62 | 05:36 | But it's on the walls where all the exciting things are happening. | ||

63 | 05:39 | The Moorish artists were denied the possibility to draw things with souls. | ||

64 | 05:43 | So they explored a more geometric art. | ||

65 | 05:45 | And so what is symmetry? | ||

66 | 05:47 | The Alhambra somehow asks all of these questions. | ||

67 | 05:50 | What is symmetry? When [there] are two of these walls, do they have the same symmetries? | ||

68 | 05:54 | Can we say whether they discovered all of the symmetries in the Alhambra? | ||

69 | 05:59 | And it was Galois who produced a language to be able to answer some of these questions. | ||

70 | 06:04 | For Galois, symmetry -- unlike for Thomas Mann, which was something still and deathly -- for Galois, symmetry was all about motion. | ||

71 | 06:12 | What can you do to a symmetrical object, move it in some way, so it looks the same as before you moved it? | ||

72 | 06:18 | I like to describe it as the magic trick moves. | ||

73 | 06:20 | What can you do to something? You close your eyes. | ||

74 | 06:22 | I do something, put it back down again. | ||

75 | 06:24 | It looks like it did before it started. | ||

76 | 06:26 | So, for example, the walls in the Alhambra -- I can take all of these tiles, and fix them at the yellow place, rotate them by 90 degrees, put them all back down again and they fit perfectly down there. | ||

77 | 06:37 | And if you open your eyes again, you wouldn't know that they'd moved. | ||

78 | 06:40 | But it's the motion that really characterizes the symmetry inside the Alhambra. | ||

79 | 06:45 | But it's also about producing a language to describe this. | ||

80 | 06:47 | And the power of mathematics is often to change one thing into another, to change geometry into language. | ||

81 | 06:54 | So I'm going to take you through, perhaps push you a little bit mathematically -- so brace yourselves -- push you a little bit to understand how this language works, which enables us to capture what is symmetry. | ||

82 | 07:04 | So, let's take these two symmetrical objects here. | ||

83 | 07:07 | Let's take the twisted six-pointed starfish. | ||

84 | 07:09 | What can I do to the starfish which makes it look the same? | ||

85 | 07:12 | Well, there I rotated it by a sixth of a turn, and still it looks like it did before I started. | ||

86 | 07:17 | I could rotate it by a third of a turn, or a half a turn, or put it back down on its image, or two thirds of a turn. | ||

87 | 07:25 | And a fifth symmetry, I can rotate it by five sixths of a turn. | ||

88 | 07:29 | And those are things that I can do to the symmetrical object that make it look like it did before I started. | ||

89 | 07:35 | Now, for Galois, there was actually a sixth symmetry. | ||

90 | 07:38 | Can anybody think what else I could do to this which would leave it like I did before I started? | ||

91 | 07:43 | I can't flip it because I've put a little twist on it, haven't I? | ||

92 | 07:46 | It's got no reflective symmetry. | ||

93 | 07:48 | But what I could do is just leave it where it is, pick it up, and put it down again. | ||

94 | 07:53 | And for Galois this was like the zeroth symmetry. | ||

95 | 07:56 | Actually, the invention of the number zero was a very modern concept, seventh century A.D., by the Indians. | ||

96 | 08:02 | It seems mad to talk about nothing. | ||

97 | 08:05 | And this is the same idea. This is a symmetrical -- so everything has symmetry, where you just leave it where it is. | ||

98 | 08:09 | So, this object has six symmetries. | ||

99 | 08:12 | And what about the triangle? | ||

100 | 08:14 | Well, I can rotate by a third of a turn clockwise or a third of a turn anticlockwise. | ||

101 | 08:20 | But now this has some reflectional symmetry. | ||

102 | 08:22 | I can reflect it in the line through X, or the line through Y, or the line through Z. Five symmetries and then of course the zeroth symmetry where I just pick it up and leave it where it is. | ||

103 | 08:34 | So both of these objects have six symmetries. | ||

104 | 08:37 | Now, I'm a great believer that mathematics is not a spectator sport, and you have to do some mathematics in order to really understand it. | ||

105 | 08:44 | So here is a little question for you. | ||

106 | 08:46 | And I'm going to give a prize at the end of my talk for the person who gets closest to the answer. | ||

107 | 08:50 | The Rubik's Cube. | ||

108 | 08:52 | How many symmetries does a Rubik's Cube have? | ||

109 | 08:55 | How many things can I do to this object and put it down so it still looks like a cube? | ||

110 | 08:59 | Okay? So I want you to think about that problem as we go on, and count how many symmetries there are. | ||

111 | 09:04 | And there will be a prize for the person who gets closest at the end. | ||

112 | 09:08 | But let's go back down to symmetries that I got for these two objects. | ||

113 | 09:12 | What Galois realized: it isn't just the individual symmetries, but how they interact with each other which really characterizes the symmetry of an object. | ||

114 | 09:21 | If I do one magic trick move followed by another, the combination is a third magic trick move. | ||

115 | 09:26 | And here we see Galois starting to develop a language to see the substance of the things unseen, the sort of abstract idea of the symmetry underlying this physical object. | ||

116 | 09:36 | For example, what if I turn the starfish by a sixth of a turn, and then a third of a turn? | ||

117 | 09:43 | So I've given names. The capital letters, A, B, C, D, E, F, are the names for the rotations. | ||

118 | 09:48 | B, for example, rotates the little yellow dot to the B on the starfish. And so on. | ||

119 | 09:54 | So what if I do B, which is a sixth of a turn, followed by C, which is a third of a turn? | ||

120 | 09:59 | Well let's do that. A sixth of a turn, followed by a third of a turn, the combined effect is as if I had just rotated it by half a turn in one go. | ||

121 | 10:08 | So the little table here records how the algebra of these symmetries work. | ||

122 | 10:13 | I do one followed by another, the answer is it's rotation D, half a turn. | ||

123 | 10:17 | What I if I did it in the other order? Would it make any difference? | ||

124 | 10:20 | Let's see. Let's do the third of the turn first, and then the sixth of a turn. | ||

125 | 10:24 | Of course, it doesn't make any difference. | ||

126 | 10:26 | It still ends up at half a turn. | ||

127 | 10:28 | And there is some symmetry here in the way the symmetries interact with each other. | ||

128 | 10:33 | But this is completely different to the symmetries of the triangle. | ||

129 | 10:36 | Let's see what happens if we do two symmetries with the triangle, one after the other. | ||

130 | 10:40 | Let's do a rotation by a third of a turn anticlockwise, and reflect in the line through X. Well, the combined effect is as if I had just done the reflection in the line through Z to start with. | ||

131 | 10:51 | Now, let's do it in a different order. | ||

132 | 10:53 | Let's do the reflection in X first, followed by the rotation by a third of a turn anticlockwise. | ||

133 | 10:59 | The combined effect, the triangle ends up somewhere completely different. | ||

134 | 11:02 | It's as if it was reflected in the line through Y. | ||

135 | 11:05 | Now it matters what order you do the operations in. | ||

136 | 11:08 | And this enables us to distinguish why the symmetries of these objects -- they both have six symmetries. So why shouldn't we say they have the same symmetries? | ||

137 | 11:16 | But the way the symmetries interact enable us -- we've now got a language to distinguish why these symmetries are fundamentally different. | ||

138 | 11:23 | And you can try this when you go down to the pub, later on. | ||

139 | 11:26 | Take a beer mat and rotate it by a quarter of a turn, then flip it. And then do it in the other order, and the picture will be facing in the opposite direction. | ||

140 | 11:35 | Now, Galois produced some laws for how these tables -- how symmetries interact. | ||

141 | 11:39 | It's almost like little Sudoku tables. | ||

142 | 11:41 | You don't see any symmetry twice in any row or column. | ||

143 | 11:45 | And, using those rules, he was able to say that there are in fact only two objects with six symmetries. | ||

144 | 11:53 | And they'll be the same as the symmetries of the triangle, or the symmetries of the six-pointed starfish. | ||

145 | 11:58 | I think this is an amazing development. | ||

146 | 12:00 | It's almost like the concept of number being developed for symmetry. | ||

147 | 12:04 | In the front here, I've got one, two, three people sitting on one, two, three chairs. | ||

148 | 12:08 | The people and the chairs are very different, but the number, the abstract idea of the number, is the same. | ||

149 | 12:14 | And we can see this now: we go back to the walls in the Alhambra. | ||

150 | 12:17 | Here are two very different walls, very different geometric pictures. | ||

151 | 12:21 | But, using the language of Galois, we can understand that the underlying abstract symmetries of these things are actually the same. | ||

152 | 12:28 | For example, let's take this beautiful wall with the triangles with a little twist on them. | ||

153 | 12:33 | You can rotate them by a sixth of a turn if you ignore the colors. We're not matching up the colors. | ||

154 | 12:37 | But the shapes match up if I rotate by a sixth of a turn around the point where all the triangles meet. | ||

155 | 12:43 | What about the center of a triangle? I can rotate by a third of a turn around the center of the triangle, and everything matches up. | ||

156 | 12:49 | And then there is an interesting place halfway along an edge, where I can rotate by 180 degrees. | ||

157 | 12:53 | And all the tiles match up again. | ||

158 | 12:56 | So rotate along halfway along the edge, and they all match up. | ||

159 | 12:59 | Now, let's move to the very different-looking wall in the Alhambra. | ||

160 | 13:03 | And we find the same symmetries here, and the same interaction. | ||

161 | 13:06 | So, there was a sixth of a turn. A third of a turn where the Z pieces meet. | ||

162 | 13:11 | And the half a turn is halfway between the six pointed stars. | ||

163 | 13:15 | And although these walls look very different, Galois has produced a language to say that in fact the symmetries underlying these are exactly the same. | ||

164 | 13:23 | And it's a symmetry we call 6-3-2. | ||

165 | 13:26 | Here is another example in the Alhambra. | ||

166 | 13:28 | This is a wall, a ceiling, and a floor. | ||

167 | 13:31 | They all look very different. But this language allows us to say that they are representations of the same symmetrical abstract object, which we call 4-4-2. Nothing to do with football, but because of the fact that there are two places where you can rotate by a quarter of a turn, and one by half a turn. | ||

168 | 13:47 | Now, this power of the language is even more, because Galois can say, "Did the Moorish artists discover all of the possible symmetries on the walls in the Alhambra?" | ||

169 | 13:56 | And it turns out they almost did. | ||

170 | 13:58 | You can prove, using Galois' language, there are actually only 17 different symmetries that you can do in the walls in the Alhambra. | ||

171 | 14:06 | And they, if you try to produce a different wall with this 18th one, it will have to have the same symmetries as one of these 17. | ||

172 | 14:14 | But these are things that we can see. | ||

173 | 14:16 | And the power of Galois' mathematical language is it also allows us to create symmetrical objects in the unseen world, beyond the two-dimensional, three-dimensional, all the way through to the four- or five- or infinite-dimensional space. | ||

174 | 14:28 | And that's where I work. I create mathematical objects, symmetrical objects, using Galois' language, in very high dimensional spaces. | ||

175 | 14:36 | So I think it's a great example of things unseen, which the power of mathematical language allows you to create. | ||

176 | 14:42 | So, like Galois, I stayed up all last night creating a new mathematical symmetrical object for you, and I've got a picture of it here. | ||

177 | 14:50 | Well, unfortunately it isn't really a picture. If I could have my board at the side here, great, excellent. | ||

178 | 14:55 | Here we are. Unfortunately, I can't show you a picture of this symmetrical object. | ||

179 | 14:59 | But here is the language which describes how the symmetries interact. | ||

180 | 15:04 | Now, this new symmetrical object does not have a name yet. | ||

181 | 15:08 | Now, people like getting their names on things, on craters on the moon or new species of animals. | ||

182 | 15:14 | So I'm going to give you the chance to get your name on a new symmetrical object which hasn't been named before. | ||

183 | 15:20 | And this thing -- species die away, and moons kind of get hit by meteors and explode -- but this mathematical object will live forever. | ||

184 | 15:27 | It will make you immortal. | ||

185 | 15:29 | In order to win this symmetrical object, what you have to do is to answer the question I asked you at the beginning. | ||

186 | 15:35 | How many symmetries does a Rubik's Cube have? | ||

187 | 15:39 | Okay, I'm going to sort you out. | ||

188 | 15:41 | Rather than you all shouting out, I want you to count how many digits there are in that number. Okay? | ||

189 | 15:46 | If you've got it as a factorial, you've got to expand the factorials. | ||

190 | 15:49 | Okay, now if you want to play, I want you to stand up, okay? | ||

191 | 15:53 | If you think you've got an estimate for how many digits, right -- we've already got one competitor here. | ||

192 | 15:58 | If you all stay down he wins it automatically. | ||

193 | 16:00 | Okay. Excellent. So we've got four here, five, six. | ||

194 | 16:03 | Great. Excellent. That should get us going. All right. | ||

195 | 16:08 | Anybody with five or less digits, you've got to sit down, because you've underestimated. | ||

196 | 16:13 | Five or less digits. So, if you're in the tens of thousands you've got to sit down. | ||

197 | 16:17 | 60 digits or more, you've got to sit down. | ||

198 | 16:20 | You've overestimated. | ||

199 | 16:22 | 20 digits or less, sit down. | ||

200 | 16:26 | How many digits are there in your number? | ||

201 | 16:31 | Two? So you should have sat down earlier. | ||

202 | 16:33 | (Laughter) Let's have the other ones, who sat down during the 20, up again. Okay? | ||

203 | 16:38 | If I told you 20 or less, stand up. | ||

204 | 16:40 | Because this one. I think there were a few here. | ||

205 | 16:42 | The people who just last sat down. | ||

206 | 16:45 | Okay, how many digits do you have in your number? | ||

207 | 16:50 | (Laughs) 21. Okay good. How many do you have in yours? | ||

208 | 16:55 | 18. So it goes to this lady here. | ||

209 | 16:58 | 21 is the closest. | ||

210 | 17:00 | It actually has -- the number of symmetries in the Rubik's cube has 25 digits. | ||

211 | 17:04 | So now I need to name this object. | ||

212 | 17:06 | So, what is your name? | ||

213 | 17:08 | I need your surname. Symmetrical objects generally -- spell it for me. | ||

214 | 17:13 | G-H-E-Z No, SO2 has already been used, actually, in the mathematical language. So you can't have that one. | ||

215 | 17:24 | So Ghez, there we go. That's your new symmetrical object. | ||

216 | 17:26 | You are now immortal. | ||

217 | 17:28 | (Applause) | ||

218 | 17:34 | And if you'd like your own symmetrical object, I have a project raising money for a charity in Guatemala, where I will stay up all night and devise an object for you, for a donation to this charity to help kids get into education in Guatemala. | ||

219 | 17:46 | And I think what drives me, as a mathematician, are those things which are not seen, the things that we haven't discovered. | ||

220 | 17:53 | It's all the unanswered questions which make mathematics a living subject. | ||

221 | 17:57 | And I will always come back to this quote from the Japanese "Essays in Idleness": "In everything, uniformity is undesirable. | ||

222 | 18:03 | Leaving something incomplete makes it interesting, and gives one the feeling that there is room for growth." Thank you. | ||

223 | 18:09 | (Applause) |