TCSE: Ted Corpus Search Engine

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


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)
3	00:28	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.
4	00:40	The young man's name was Evariste Galois.
5	00:43	He was a well-known revolutionary in Paris at the time.
6	00:47	Galois was taken to the local hospital where he died the next day in the arms of his brother.
7	00:53	And the last words he said to his brother were, "Don't cry for me, Alfred.
8	00:57	I need all the courage I can muster to die at the age of 20."
9	01:03	It wasn't, in fact, revolutionary politics for which Galois was famous.
10	01:07	But a few years earlier, while still at school, he'd actually cracked one of the big mathematical problems at the time.
11	01:14	And he wrote to the academicians in Paris, trying to explain his theory.
12	01:18	But the academicians couldn't understand anything that he wrote.
13	01:21	(Laughter)
14	01:22	This is how he wrote most of his mathematics.
15	01:25	So, the night before that duel, he realized this possibly is his last chance to try and explain his great breakthrough.
16	01:32	So he stayed up the whole night, writing away, trying to explain his ideas.
17	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.
18	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.
19	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.
20	02:00	Now, symmetry is almost nature's language.
21	02:02	It helps us to understand so many different bits of the scientific world.
22	02:06	For example, molecular structure.
23	02:08	What crystals are possible, we can understand through the mathematics of symmetry.
24	02:14	In microbiology you really don't want to get a symmetrical object, because they are generally rather nasty.
25	02:18	The swine flu virus, at the moment, is a symmetrical object.
26	02:21	And it uses the efficiency of symmetry to be able to propagate itself so well.
27	02:27	But on a larger scale of biology, actually symmetry is very important, because it actually communicates genetic information.
28	02:32	I've taken two pictures here and I've made them artificially symmetrical.
29	02:36	And if I ask you which of these you find more beautiful, you're probably drawn to the lower two.
30	02:41	Because it is hard to make symmetry.
31	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.
32	02:51	So symmetry is a language which can help to communicate genetic information.
33	02:56	Symmetry can also help us to explain what's happening in the Large Hadron Collider in CERN.
34	03:01	Or what's not happening in the Large Hadron Collider in CERN.
35	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.
36	03:14	And I think Galileo summed up, very nicely, the power of mathematics to understand the scientific world around us.
37	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.
38	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."
39	03:38	But it's not just scientists who are interested in symmetry.
40	03:41	Artists too love to play around with symmetry.
41	03:44	They also have a slightly more ambiguous relationship with it.
42	03:47	Here is Thomas Mann talking about symmetry in "The Magic Mountain."
43	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."
44	03:59	But what artists like to do is to set up expectations of symmetry and then break them.
45	04:03	And a beautiful example of this I found, actually, when I visited a colleague of mine in Japan, Professor Kurokawa.
46	04:09	And he took me up to the temples in Nikko.
47	04:12	And just after this photo was taken we walked up the stairs.
48	04:15	And the gateway you see behind has eight columns, with beautiful symmetrical designs on them.
49	04:20	Seven of them are exactly the same, and the eighth one is turned upside down.
50	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."
51	04:32	And he said, "No, no, no. It was a very deliberate act."
52	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.
53	04:45	Leaving something incomplete makes it interesting, and gives one the feeling that there is room for growth."
54	04:50	Even when building the Imperial Palace, they always leave one place unfinished.
55	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.
56	05:06	This is a palace celebrating symmetry.
57	05:08	Recently I took my family -- we do these rather kind of nerdy mathematical trips, which my family love.
58	05:13	This is my son Tamer. You can see he's really enjoying our mathematical trip to the Alhambra.
59	05:18	But I wanted to try and enrich him.
60	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.
61	05:27	So, I wanted to open his eyes up to how much symmetry is running through the Alhambra.
62	05:32	You see it already. Immediately you go in, the reflective symmetry in the water.
63	05:36	But it's on the walls where all the exciting things are happening.
64	05:39	The Moorish artists were denied the possibility to draw things with souls.
65	05:43	So they explored a more geometric art.
66	05:45	And so what is symmetry?
67	05:47	The Alhambra somehow asks all of these questions.
68	05:50	What is symmetry? When [there] are two of these walls, do they have the same symmetries?
69	05:54	Can we say whether they discovered all of the symmetries in the Alhambra?
70	05:59	And it was Galois who produced a language to be able to answer some of these questions.
71	06:04	For Galois, symmetry -- unlike for Thomas Mann, which was something still and deathly -- for Galois, symmetry was all about motion.
72	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?
73	06:18	I like to describe it as the magic trick moves.
74	06:20	What can you do to something? You close your eyes.
75	06:22	I do something, put it back down again.
76	06:24	It looks like it did before it started.
77	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.
78	06:37	And if you open your eyes again, you wouldn't know that they'd moved.
79	06:40	But it's the motion that really characterizes the symmetry inside the Alhambra.
80	06:45	But it's also about producing a language to describe this.
81	06:47	And the power of mathematics is often to change one thing into another, to change geometry into language.
82	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.
83	07:04	So, let's take these two symmetrical objects here.
84	07:07	Let's take the twisted six-pointed starfish.
85	07:09	What can I do to the starfish which makes it look the same?
86	07:12	Well, there I rotated it by a sixth of a turn, and still it looks like it did before I started.
87	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.
88	07:25	And a fifth symmetry, I can rotate it by five sixths of a turn.
89	07:29	And those are things that I can do to the symmetrical object that make it look like it did before I started.
90	07:35	Now, for Galois, there was actually a sixth symmetry.
91	07:38	Can anybody think what else I could do to this which would leave it like I did before I started?
92	07:43	I can't flip it because I've put a little twist on it, haven't I?
93	07:46	It's got no reflective symmetry.
94	07:48	But what I could do is just leave it where it is, pick it up, and put it down again.
95	07:53	And for Galois this was like the zeroth symmetry.
96	07:56	Actually, the invention of the number zero was a very modern concept, seventh century A.D., by the Indians.
97	08:02	It seems mad to talk about nothing.
98	08:05	And this is the same idea. This is a symmetrical -- so everything has symmetry, where you just leave it where it is.
99	08:09	So, this object has six symmetries.
100	08:12	And what about the triangle?
101	08:14	Well, I can rotate by a third of a turn clockwise or a third of a turn anticlockwise.
102	08:20	But now this has some reflectional symmetry.
103	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.
104	08:34	So both of these objects have six symmetries.
105	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.
106	08:44	So here is a little question for you.
107	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.
108	08:50	The Rubik's Cube.
109	08:52	How many symmetries does a Rubik's Cube have?
110	08:55	How many things can I do to this object and put it down so it still looks like a cube?
111	08:59	Okay? So I want you to think about that problem as we go on, and count how many symmetries there are.
112	09:04	And there will be a prize for the person who gets closest at the end.
113	09:08	But let's go back down to symmetries that I got for these two objects.
114	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.
115	09:21	If I do one magic trick move followed by another, the combination is a third magic trick move.
116	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.
117	09:36	For example, what if I turn the starfish by a sixth of a turn, and then a third of a turn?
118	09:43	So I've given names. The capital letters, A, B, C, D, E, F, are the names for the rotations.
119	09:48	B, for example, rotates the little yellow dot to the B on the starfish. And so on.
120	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?
121	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.
122	10:08	So the little table here records how the algebra of these symmetries work.
123	10:13	I do one followed by another, the answer is it's rotation D, half a turn.
124	10:17	What I if I did it in the other order? Would it make any difference?
125	10:20	Let's see. Let's do the third of the turn first, and then the sixth of a turn.
126	10:24	Of course, it doesn't make any difference.
127	10:26	It still ends up at half a turn.
128	10:28	And there is some symmetry here in the way the symmetries interact with each other.
129	10:33	But this is completely different to the symmetries of the triangle.
130	10:36	Let's see what happens if we do two symmetries with the triangle, one after the other.
131	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.
132	10:51	Now, let's do it in a different order.
133	10:53	Let's do the reflection in X first, followed by the rotation by a third of a turn anticlockwise.
134	10:59	The combined effect, the triangle ends up somewhere completely different.
135	11:02	It's as if it was reflected in the line through Y.
136	11:05	Now it matters what order you do the operations in.
137	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?
138	11:16	But the way the symmetries interact enable us -- we've now got a language to distinguish why these symmetries are fundamentally different.
139	11:23	And you can try this when you go down to the pub, later on.
140	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.
141	11:35	Now, Galois produced some laws for how these tables -- how symmetries interact.
142	11:39	It's almost like little Sudoku tables.
143	11:41	You don't see any symmetry twice in any row or column.
144	11:45	And, using those rules, he was able to say that there are in fact only two objects with six symmetries.
145	11:53	And they'll be the same as the symmetries of the triangle, or the symmetries of the six-pointed starfish.
146	11:58	I think this is an amazing development.
147	12:00	It's almost like the concept of number being developed for symmetry.
148	12:04	In the front here, I've got one, two, three people sitting on one, two, three chairs.
149	12:08	The people and the chairs are very different, but the number, the abstract idea of the number, is the same.
150	12:14	And we can see this now: we go back to the walls in the Alhambra.
151	12:17	Here are two very different walls, very different geometric pictures.
152	12:21	But, using the language of Galois, we can understand that the underlying abstract symmetries of these things are actually the same.
153	12:28	For example, let's take this beautiful wall with the triangles with a little twist on them.
154	12:33	You can rotate them by a sixth of a turn if you ignore the colors. We're not matching up the colors.
155	12:37	But the shapes match up if I rotate by a sixth of a turn around the point where all the triangles meet.
156	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.
157	12:49	And then there is an interesting place halfway along an edge, where I can rotate by 180 degrees.
158	12:53	And all the tiles match up again.
159	12:56	So rotate along halfway along the edge, and they all match up.
160	12:59	Now, let's move to the very different-looking wall in the Alhambra.
161	13:03	And we find the same symmetries here, and the same interaction.
162	13:06	So, there was a sixth of a turn. A third of a turn where the Z pieces meet.
163	13:11	And the half a turn is halfway between the six pointed stars.
164	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.
165	13:23	And it's a symmetry we call 6-3-2.
166	13:26	Here is another example in the Alhambra.
167	13:28	This is a wall, a ceiling, and a floor.
168	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.
169	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?"
170	13:56	And it turns out they almost did.
171	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.
172	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.
173	14:14	But these are things that we can see.
174	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.
175	14:28	And that's where I work. I create mathematical objects, symmetrical objects, using Galois' language, in very high dimensional spaces.
176	14:36	So I think it's a great example of things unseen, which the power of mathematical language allows you to create.
177	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.
178	14:50	Well, unfortunately it isn't really a picture. If I could have my board at the side here, great, excellent.
179	14:55	Here we are. Unfortunately, I can't show you a picture of this symmetrical object.
180	14:59	But here is the language which describes how the symmetries interact.
181	15:04	Now, this new symmetrical object does not have a name yet.
182	15:08	Now, people like getting their names on things, on craters on the moon or new species of animals.
183	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.
184	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.
185	15:27	It will make you immortal.
186	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.
187	15:35	How many symmetries does a Rubik's Cube have?
188	15:39	Okay, I'm going to sort you out.
189	15:41	Rather than you all shouting out, I want you to count how many digits there are in that number. Okay?
190	15:46	If you've got it as a factorial, you've got to expand the factorials.
191	15:49	Okay, now if you want to play, I want you to stand up, okay?
192	15:53	If you think you've got an estimate for how many digits, right -- we've already got one competitor here.
193	15:58	If you all stay down he wins it automatically.
194	16:00	Okay. Excellent. So we've got four here, five, six.
195	16:03	Great. Excellent. That should get us going. All right.
196	16:08	Anybody with five or less digits, you've got to sit down, because you've underestimated.
197	16:13	Five or less digits. So, if you're in the tens of thousands you've got to sit down.
198	16:17	60 digits or more, you've got to sit down.
199	16:20	You've overestimated.
200	16:22	20 digits or less, sit down.
201	16:26	How many digits are there in your number?
202	16:31	Two? So you should have sat down earlier.
203	16:33	(Laughter)
204	16:34	Let's have the other ones, who sat down during the 20, up again. Okay?
205	16:38	If I told you 20 or less, stand up.
206	16:40	Because this one. I think there were a few here.
207	16:42	The people who just last sat down.
208	16:45	Okay, how many digits do you have in your number?
209	16:50	(Laughs) 21. Okay good. How many do you have in yours?
210	16:55	18. So it goes to this lady here.
211	16:58	21 is the closest.
212	17:00	It actually has -- the number of symmetries in the Rubik's cube has 25 digits.
213	17:04	So now I need to name this object.
214	17:06	So, what is your name?
215	17:08	I need your surname. Symmetrical objects generally -- spell it for me.
216	17:13	G-H-E-Z No, SO2 has already been used, actually, in the mathematical language. So you can't have that one.
217	17:24	So Ghez, there we go. That's your new symmetrical object.
218	17:26	You are now immortal.
219	17:28	(Applause)
220	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.
221	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.
222	17:53	It's all the unanswered questions which make mathematics a living subject.
223	17:57	And I will always come back to this quote from the Japanese "Essays in Idleness": "In everything, uniformity is undesirable.
224	18:03	Leaving something incomplete makes it interesting, and gives one the feeling that there is room for growth." Thank you.
225	18:09	(Applause)

Marcus du Sautoy: Symmetry, reality's riddle