1
00:00:18,040 --> 00:00:25,940
Well, the 30th of May 1832, a gunshot was heard ringing out across the building this morning.

2
00:00:25,960 --> 00:00:30,640
Paris, a peasant who was walking to the market that morning, ran towards the end.

3
00:00:30,640 --> 00:00:38,170
Gunshot was over and on the ground, he found a young man writhing in pain, clearly shot by a shooting.

4
00:00:39,250 --> 00:00:46,600
The young man was taken to a local hospital in a country hospital where he to stay in the arms of his brother, Alfred.

5
00:00:46,690 --> 00:00:50,110
The lawsuits, he says, were too high. Me, Alfred.

6
00:00:50,200 --> 00:00:54,850
I need all the courage I can muster to die at the age of 20.

7
00:00:55,840 --> 00:01:02,950
The young man's name was every Galois. He was a well-known revolutionary in Paris at the time.

8
00:01:04,810 --> 00:01:08,740
Actually, this is if you want to see me any private at all.

9
00:01:10,700 --> 00:01:19,720
Yes. Yes. Now, this is that is actually happening when I was about ten years before the IDF came out and.

10
00:01:22,300 --> 00:01:26,300
The young man. The young man's name was every schoolboy.

11
00:01:26,560 --> 00:01:29,620
He was a well-known revolutionary in Paris at the time.

12
00:01:30,130 --> 00:01:34,600
Now, it's not quite clear what the jewel that morning was about.

13
00:01:35,440 --> 00:01:39,650
Some say that it was, in fact, a duel with a friend of his over a woman.

14
00:01:39,730 --> 00:01:48,070
They both had fallen in love with others. And actually it was the establishment trying to get rid of this troublesome revolutionary.

15
00:01:48,730 --> 00:01:55,510
There's even a conjecture that he staged his own dance in order to try and spark a new revolution in Paris.

16
00:01:56,080 --> 00:02:04,180
Well, now it's unclear what that duel was about, but it wasn't revolutionary politics for which this young man became famous,

17
00:02:04,180 --> 00:02:13,600
because a few years earlier, while still at school, every school law had solved one of the big mathematical problems at the time.

18
00:02:14,410 --> 00:02:18,700
Now, he went to the parents academy trying to explain his great breakthrough.

19
00:02:18,940 --> 00:02:27,130
But the academicians in Paris couldn't understand a word of what he was talking about, partly because this is how he wrote most of his mathematics.

20
00:02:28,980 --> 00:02:29,300
Anyway,

21
00:02:29,530 --> 00:02:38,320
the night before that duel he realised that probably this was going to be his last chance to try and explain the great breakthrough that he made.

22
00:02:38,350 --> 00:02:47,110
So he actually set off the whole night before the duel, writing the line to a friend, trying to articulate this great breakthrough he made.

23
00:02:48,460 --> 00:02:52,840
And as a dawn arose and the new day, he went out to meet his destiny.

24
00:02:53,470 --> 00:02:57,750
Probably in fact been doing maths all night with the fact that he was such a bad shot that morning.

25
00:02:57,760 --> 00:03:03,030
But. But as you say, there's documents that he left behind on his desk.

26
00:03:03,040 --> 00:03:10,989
Many of us regard as some of the most important mathematical documents in the whole history of my subjects, because hidden in so many,

27
00:03:10,990 --> 00:03:23,140
that was a new language, a language to help us to explain one of the biggest ideas in mathematics and in science, namely the idea of symmetry.

28
00:03:23,500 --> 00:03:26,620
This language that he developed was called group theory.

29
00:03:26,920 --> 00:03:35,140
Now, I, as a mathematician, use this language that got a lot of evidence already, just by the age of 20,

30
00:03:35,320 --> 00:03:41,770
every day in my working life, in order to be able to understand and navigate the world of symmetry.

31
00:03:42,130 --> 00:03:48,460
It's interesting. Although I became a mathematician on my Galois, I think our Morning Edition went on.

32
00:03:48,700 --> 00:03:52,210
When I was younger, I hadn't wanted to be a mathematician at all.

33
00:03:52,240 --> 00:03:55,600
In fact, my dream when I was a school was in fact to become a spy.

34
00:03:57,100 --> 00:04:04,660
This is partly fuelled by my mother, who works for the Foreign Office, and when she had children, she had to retire.

35
00:04:05,230 --> 00:04:10,510
That's a she told me and my sister that she'd been allowed to keep the black gun of

36
00:04:10,510 --> 00:04:15,640
every member of the Foreign Office and that it was hidden somewhere inside our house.

37
00:04:16,180 --> 00:04:22,030
So my sister and I used to spend all our face and try to find where this black gun was.

38
00:04:22,270 --> 00:04:27,010
Never find it. They obviously told her the answer concealed very well as well, but some.

39
00:04:27,220 --> 00:04:31,180
So I decided that I was going to join the Foreign Office like my mother.

40
00:04:31,180 --> 00:04:39,970
And then I thought having becoming a spy. So when I got to secondary school, I went to a safe school here in Oxfordshire.

41
00:04:41,020 --> 00:04:50,110
I signed on for every language that my my school, the Foreign Office, found any languages, the French, Latin, German.

42
00:04:50,680 --> 00:04:56,379
Actually, at the time when I went to school, there was a course on the BBC teaching Russian and I thought,

43
00:04:56,380 --> 00:05:03,450
Oh, fantastic, that's what you need for a spy. Now you can tell how old I am because I grew up in the Cold War when the Russian fabric.

44
00:05:04,750 --> 00:05:09,340
And so my French teacher help me with the Russian course on the BBC.

45
00:05:10,060 --> 00:05:12,010
But as I wanted to learn these languages,

46
00:05:12,010 --> 00:05:18,500
I more and more frustrated because there were all these kind of strange spellings that you just have to learn.

47
00:05:18,520 --> 00:05:25,329
It was never long term at Learn Irregular Verbs, which is S.A. Cotton to the Russian course was actually a disaster.

48
00:05:25,330 --> 00:05:29,490
I couldn't get past the word hello, which has so many consonants and no vowels.

49
00:05:30,070 --> 00:05:34,890
My strong suit was the name of the cause. I couldn't even say the name of the school said.

50
00:05:35,610 --> 00:05:44,679
So I became very disillusioned. And it was strange to kind of spy and learn all these languages, but it was about that time, about 12 or 13,

51
00:05:44,680 --> 00:05:53,020
that my maths teacher in my school in May, Wilson, just like I want to see you after class, I got into trouble.

52
00:05:53,770 --> 00:05:58,870
So I went off to the end, of course ended and he said, Follow me.

53
00:05:59,050 --> 00:06:05,620
We went round the back of my small car, the whole dropping out, and then he took out his right time.

54
00:06:05,620 --> 00:06:09,100
Cigar. You said you weren't allowed to smoke in the common room. So two guys, right?

55
00:06:09,310 --> 00:06:17,379
So I said the same time, I think you should find out what mathematics is really about because it isn't really about all the kind of multiplication,

56
00:06:17,380 --> 00:06:20,550
percentages, signs and cosines that we're doing in the classroom.

57
00:06:20,800 --> 00:06:24,930
Much more exciting and he recommended a few books to me.

58
00:06:25,360 --> 00:06:32,439
But he's always a rock, this world of mathematics. And so we took this I took this list of books,

59
00:06:32,440 --> 00:06:39,670
and I went home and my dad and I came up and we came to Oxford with this wonderful bookshop called Blackwell's,

60
00:06:40,240 --> 00:06:47,320
which is going to have, you know, your daughter, who, as you know, I think that was taught, is this kind of small shopfront.

61
00:06:47,590 --> 00:06:51,250
Do you think there can't be anything interesting here? You go inside and it just goes on forever.

62
00:06:52,010 --> 00:06:56,829
You go down into the basement and there are all these science books and maths books, and that was so exciting.

63
00:06:56,830 --> 00:07:02,580
And so my dad took this list of books and I was just went round sort of wandering, putting books off the shelves.

64
00:07:02,610 --> 00:07:09,790
I couldn't understand words. You seem to be in some sort of secret code, but there are undergraduate stands of reading up against the bookshelves,

65
00:07:09,790 --> 00:07:13,099
reading these books as if they were novels, little stories and things.

66
00:07:13,100 --> 00:07:16,570
So. So I became very intrigued. It was exciting excitement so much.

67
00:07:16,900 --> 00:07:23,080
And we took these books home and I began to read about this exciting world of mathematics,

68
00:07:23,080 --> 00:07:27,940
and I began to have a sense of the amazing stories that this world contains.

69
00:07:28,330 --> 00:07:35,290
But what intrigued Moses was one of the books that my teacher recommended because it was called The Language of Mathematics.

70
00:07:35,710 --> 00:07:39,370
I still have the book that we bought that Saturday morning.

71
00:07:39,370 --> 00:07:46,060
It costs £1.75. I defy you to find a book in black holes now that costs £1.79.

72
00:07:46,720 --> 00:07:53,170
But this book I know because first of all, I never really thought of mathematics as a language,

73
00:07:55,060 --> 00:07:58,840
but as I began to read this book, I realised, Yeah, it's an amazing language.

74
00:07:58,840 --> 00:08:06,000
It's a language which actually helps us to articulate the structures in the world around us, where we come from, where we're going to go next.

75
00:08:06,460 --> 00:08:11,100
It was one language because I didn't have the irregular verbs that only make perfect sense.

76
00:08:11,110 --> 00:08:18,939
And so this is the language for me. So that's not to say that they didn't have strange twists and turns and surprises,

77
00:08:18,940 --> 00:08:25,330
and that's one of the joys of mathematics that not everything is totally obvious, yet everything seemed to make perfect, logical sense.

78
00:08:25,720 --> 00:08:30,940
And this is the book. It's not a very famous it's been really helpful in language.

79
00:08:30,940 --> 00:08:37,389
Mathematics is a language on the language that really I found so intoxicating in this book was this

80
00:08:37,390 --> 00:08:42,820
man with power who developed to understand the world of symmetry and language through group theory.

81
00:08:43,390 --> 00:08:48,460
And I began to understand when I read these books that in a way, symmetry is its own language.

82
00:08:48,640 --> 00:08:53,560
Symmetry is a language that nature uses to communicate information.

83
00:08:54,040 --> 00:08:57,820
If something has symmetry, then it's generally got a message inside of it.

84
00:08:58,690 --> 00:09:07,419
For example, if you go to the garden and you look at the bumblebee, a bumblebee has incredibly fine vision and can't judge distances.

85
00:09:07,420 --> 00:09:14,979
It sees the world very monochromatic. KLEE But what it can make out is shapes with symmetry because it knows the shape.

86
00:09:14,980 --> 00:09:18,790
The symmetry is likely to be a flower, which will have substance inside it.

87
00:09:19,180 --> 00:09:25,180
Now, intriguingly, the flower itself needs to be to this day in order to propagate is genetic heritage.

88
00:09:25,450 --> 00:09:30,550
So the symmetry of the flower is little bit like a bill saying, you know, come and visit me.

89
00:09:31,180 --> 00:09:39,630
And in fact, research has been done to show that the more symmetrical the flower, the sweeter the nectar inside the bee, inside the flower.

90
00:09:39,640 --> 00:09:48,280
Although we also have become very sensitive to symmetry, I think, you know, we survived in the jungle because of our ability to spot things,

91
00:09:48,310 --> 00:09:55,030
especially if you're in the jungle, you see the chaos of the trees and the leaves and things, and suddenly you see something with mirror symmetry.

92
00:09:55,630 --> 00:10:00,280
You better take notice because that's an animal. Either it's going to eat you or you're going to it.

93
00:10:00,370 --> 00:10:04,330
So symmetry. Those who are sensitive to symmetry somehow survive in this world.

94
00:10:05,200 --> 00:10:09,460
And even now we use symmetry very often when we're choosing a mate.

95
00:10:10,030 --> 00:10:15,880
I'm going to show you two pictures of faces I talk to, just think normally, but I made them artificially symmetrical.

96
00:10:16,120 --> 00:10:23,410
And the bottom line now, if you ask somebody between these two images, which ones do you find the most beautiful?

97
00:10:23,770 --> 00:10:27,430
Most people are drawn to the face, which is most symmetrical.

98
00:10:27,970 --> 00:10:31,690
And why is that? Why we associate symmetry with beauty?

99
00:10:31,900 --> 00:10:35,590
Well, again, symmetry is communicating information. We're looking for a mate.

100
00:10:35,740 --> 00:10:41,050
We're looking for somebody who has very good genetic heritage, good upbringing, good conditions.

101
00:10:41,470 --> 00:10:45,340
And the thing is that symmetry is quite hard to make. It's very easy to break.

102
00:10:45,640 --> 00:10:50,890
And so if you find somebody who has a perfectly symmetrical face that's actually communicating information,

103
00:10:51,250 --> 00:10:55,690
it's communicating information about their genetic background, that they're their upbringing.

104
00:10:55,690 --> 00:11:01,810
And so that's why, again, symmetry is being used in the face to communicate information when you're choosing your mate.

105
00:11:02,650 --> 00:11:05,350
And I rather like quite which comes from Galileo,

106
00:11:05,350 --> 00:11:11,380
which I think rather sums up the power of mathematics as a language to to understand the world around us.

107
00:11:11,440 --> 00:11:18,070
He writes, The universe cannot be raised until we learn the language, become familiar with the characters, which is written.

108
00:11:18,760 --> 00:11:20,650
It is written in mathematical language.

109
00:11:21,040 --> 00:11:29,380
And the letters are triangles, circles and other geometric figures without which means it is humanly impossible to comprehend a single without.

110
00:11:31,220 --> 00:11:38,000
So where do we start to try and explore the world as soon as we go off to the beginning of the 19th century,

111
00:11:38,000 --> 00:11:44,990
developed this language, but actually maybe trying to articulate not a sad world of symmetry for millennia.

112
00:11:45,200 --> 00:11:50,370
In fact, if you go back to some of the very first objects that humans carved out.

113
00:11:50,390 --> 00:11:56,480
So there's a wonderful game in which we find in the British Museum in London.

114
00:11:57,170 --> 00:12:04,129
This game dates back to 2600 B.C. and it's got for the first time, interestingly,

115
00:12:04,130 --> 00:12:10,340
the first ice cubes that we used to play, Monopoly, things like that, but a little tetrahedral ice.

116
00:12:10,910 --> 00:12:19,280
So it's a kindred symmetrical object. The first choice that we used in history were the knuckle bones of sheep, which all are irregular.

117
00:12:19,280 --> 00:12:22,490
They would land. Several are often on one side rather than another.

118
00:12:22,700 --> 00:12:28,790
And people soon realised that if you wanted to find ice, you want something which is symmetrical, which doesn't on one side over the other.

119
00:12:28,970 --> 00:12:34,540
So they were naturally drawn by their claim to produce shapes which had an inbuilt symmetry.

120
00:12:35,060 --> 00:12:42,950
And so the four sides of the novel gradually got carved into this tetrahedron, which is made up of four equal outlines.

121
00:12:43,130 --> 00:12:46,610
Now, interestingly, I like this one and one of the points upwards.

122
00:12:46,880 --> 00:12:55,430
So what's the the they describe this game as they would colour two corners and then you throw it the tetrahedron

123
00:12:56,030 --> 00:13:00,290
and then you count the number of black spots that were pointing out and that would be all new in the game.

124
00:13:00,830 --> 00:13:04,850
And so this game of war actually is one of the forerunners of backgammon.

125
00:13:04,850 --> 00:13:06,500
It's an early version of backgammon.

126
00:13:06,770 --> 00:13:15,649
And even the board itself, you can see some actually the significant squares all have symmetrical shapes attached to them around the same time.

127
00:13:15,650 --> 00:13:19,670
But in Scotland you find other artists playing around with symmetry.

128
00:13:20,030 --> 00:13:25,280
In fact, these stone balls you couldn't find here in Oxford in the Ashmolean Museum,

129
00:13:26,000 --> 00:13:35,420
they make they match 2500 B.C. Every game you can see the artist taking a ball and carving different patches on the side of the ball,

130
00:13:35,420 --> 00:13:38,930
almost like a football, trying to make out of all the different patches.

131
00:13:39,170 --> 00:13:42,680
And you can see the art is exploring different possibilities.

132
00:13:42,680 --> 00:13:48,020
So something like a cube or the tetrahedron or some more complicated shapes.

133
00:13:48,380 --> 00:13:53,510
And it's interesting, we don't actually know what these stones or whether they had a purpose.

134
00:13:53,930 --> 00:13:55,460
They don't seem to be part of the game.

135
00:13:56,270 --> 00:14:03,830
Some have suggested they might have been part of a fortune telling, but it isn't the way they had power for the clans.

136
00:14:04,340 --> 00:14:11,690
But it's facilitating all these 2500 B.C., exploring one possible symmetries that are out there.

137
00:14:11,810 --> 00:14:19,040
And that's somehow what drives mathematics, is that the the the challenge of trying to find what possible symmetries are.

138
00:14:19,700 --> 00:14:27,590
And it really was it wasn't we had to wait until the ancient Greeks for a more analytical analysis of these symmetrical shapes began to emerge.

139
00:14:28,220 --> 00:14:34,730
And in fact, you find in Euclid's elements a proof sort of culmination of this great book.

140
00:14:34,970 --> 00:14:40,760
Is it a proof that you can actually only have five different shapes which make up dice?

141
00:14:41,060 --> 00:14:47,330
It always has a symmetrical face, and you have to have the same symmetrical face about the whole shape.

142
00:14:47,330 --> 00:14:52,640
And they have to arrange such that no faces and this can be distinguished from any other.

143
00:14:53,120 --> 00:14:59,470
With those conditions you find there are only five different possible types and actually place.

144
00:14:59,540 --> 00:15:06,529
I believe that the symmetry so important is that the associated these five basic shapes with the building

145
00:15:06,530 --> 00:15:11,900
blocks of nature we believe in mathematics was the foundation of the whole of the natural world.

146
00:15:12,170 --> 00:15:14,870
And so each of these five shapes is associated with one them.

147
00:15:14,870 --> 00:15:20,450
One of the elements now the elements at of time in the ancient Greeks was things like oxygen and hydrogen,

148
00:15:21,140 --> 00:15:24,260
but with things like fire, earth, air and water.

149
00:15:24,860 --> 00:15:28,189
So fire, the shape of fire was this little tetrahedron.

150
00:15:28,190 --> 00:15:32,179
So the tetrahedron is sometimes spiky, is of all the dice.

151
00:15:32,180 --> 00:15:39,650
And that was the shape of fire. The cube made up of its six square faces that was stable earth.

152
00:15:40,100 --> 00:15:46,610
And then you have another shape made out of rats and triangles. You could put eight equal out to times together to make an octahedron.

153
00:15:47,180 --> 00:15:55,460
That was the shape of this place. As I say to air, you can put 20 faces together, an easy classify move to make an icosahedron.

154
00:15:55,940 --> 00:16:00,500
This is the most circled of the shapes and so Plato associated with water.

155
00:16:00,770 --> 00:16:02,600
And then you had this one shape match over,

156
00:16:03,080 --> 00:16:11,270
which was the dodecahedron beautiful shape made 212 pentagonal faces and he associated this one with the shape of the universe.

157
00:16:11,540 --> 00:16:19,239
At the time, the ancient Greeks believe that somehow the universe was basically just some sort of glass ball with the star painted on it and place.

158
00:16:19,240 --> 00:16:27,229
They believe that that's where we were sitting. Inside was basically just a big dodecahedron and you can find examples.

159
00:16:27,230 --> 00:16:36,260
Actually, again, it's 12 faces. There are no examples of called calendars or with Zodiac signs with these 12 phases.

160
00:16:36,690 --> 00:16:38,030
So, yes, it's interesting.

161
00:16:38,900 --> 00:16:47,690
Now, from a small world perspective, Plato's classification of these shapes associated with the elements as we find what looks kind of ridiculous.

162
00:16:48,020 --> 00:16:52,110
But actually, Plato talked to something very deep about symmetry.

163
00:16:52,250 --> 00:16:59,420
So there was this symmetry. There is symmetry absolutely at the heart of science and the natural world.

164
00:16:59,990 --> 00:17:08,050
So, for example, in chemistry, chemists learn about different ways that molecules can be put together as way.

165
00:17:08,840 --> 00:17:16,970
So the different credit, the way that crystals behave of crystal structures, is very often dependent on proteins or symmetry.

166
00:17:17,540 --> 00:17:23,820
The biologies also in biology don't really want to guess because nobody knows.

167
00:17:24,290 --> 00:17:29,420
Generally, viruses like herpes virus or the AIDS virus based on symmetrical shapes.

168
00:17:29,690 --> 00:17:36,319
In fact, that virus is using the fact that symmetry has a very simple rule to create the object.

169
00:17:36,320 --> 00:17:45,590
It's not a very complicated object. So virus is heart broken has a very short is of RNA or DNA, which is a program to reproduce itself.

170
00:17:45,620 --> 00:17:48,020
It needs to reproduce itself as quickly as possible.

171
00:17:48,410 --> 00:17:54,059
And so the fact that a symmetrical shape is a very false shape to make with the very simple program is probably at

172
00:17:54,060 --> 00:18:02,480
the heart of why the virus is choosing to use this sort of a very stable shape and a very strong shape as well.

173
00:18:03,110 --> 00:18:09,800
And so we've begun to understand the way that viruses work. Understanding online symmetry in physics as well.

174
00:18:09,890 --> 00:18:13,160
And what's happening in a Large Hadron Collider,

175
00:18:13,490 --> 00:18:18,319
the fact that we're able to predict new particles that combined in there to make

176
00:18:18,320 --> 00:18:22,160
sense of all this kind of menagerie of particles that we've discovered so far.

177
00:18:22,430 --> 00:18:30,470
Actually, this horse is a simple object in very high dimensions, which somehow makes sense of all of these strange objects that we discovered.

178
00:18:30,890 --> 00:18:34,220
And also that helps us to to make predictions about what's missing.

179
00:18:34,430 --> 00:18:40,580
Because if one of the pieces isn't represented, there's likely to be a particle and we can look for it will correspond to that face.

180
00:18:40,880 --> 00:18:48,350
So symmetry is all over the scientific world, but essentially symmetry is very much part of the artistic as well,

181
00:18:48,890 --> 00:18:53,960
although I think all of these have a slightly more ambiguous relationship with symmetry.

182
00:18:54,530 --> 00:19:03,410
This is Thomas Mann in the Magic Mountain, talking about the symmetry of a snowflake, and they might be a shot at precision.

183
00:19:03,690 --> 00:19:11,120
And it definitely the very marrow of death, I think for all this symmetry is something they're a little bit nervous of.

184
00:19:11,120 --> 00:19:12,259
It's very restrictive.

185
00:19:12,260 --> 00:19:22,460
It tells you what should happen and and just like how it somehow can be filled with a sense of something without life and energy to it.

186
00:19:22,850 --> 00:19:26,419
But I think there are some artists who really love revelling in symmetry.

187
00:19:26,420 --> 00:19:33,500
And one of the parts, and particularly taps into symmetry and fantastic I think is music and in particular the music of ball.

188
00:19:33,530 --> 00:19:39,630
For example, the Goldberg Variations, I think is almost like a song celebrating symmetry.

189
00:19:39,640 --> 00:19:43,000
So here's a little bit of one of the variations from Bach, Soledad.

190
00:19:43,040 --> 00:20:02,699
Variations. Now the variations actually start with an aria, very simple aria.

191
00:20:02,700 --> 00:20:09,150
And then there are three variations through to that all study, which is the reputation of the aria you saw.

192
00:20:10,050 --> 00:20:15,810
Now, interestingly, so there's sort of a circle going on there already because the two actually got one means to the other.

193
00:20:16,020 --> 00:20:22,379
And you go to the 16th variation, Bach pulls out a banjo, which usually is the beginning of a musical piece.

194
00:20:22,380 --> 00:20:25,100
So it really got the sense with this piece.

195
00:20:25,350 --> 00:20:34,079
So all four ends and there's a sense, a circle already there that everything that variation Bach makes a hand out of the variation.

196
00:20:34,080 --> 00:20:39,000
So a kind of you might remember from singing in school when all this all starts off the song.

197
00:20:39,000 --> 00:20:42,479
And then a little bit later, the other half of the class joins in.

198
00:20:42,480 --> 00:20:48,090
And you have to tune and the same tune, guys, with the different times and you get this lovely interaction.

199
00:20:48,360 --> 00:20:52,700
So Bach loved the idea of canons, but he wanted something slightly more sophisticated.

200
00:20:52,710 --> 00:20:58,200
So each new canon, actually the second voice when it comes in doesn't just repeat with the first,

201
00:20:59,100 --> 00:21:03,060
but I think the set up was so soft, higher and higher and higher.

202
00:21:03,240 --> 00:21:09,110
So actually, by the eighth variation, what happens is that you guess the second voice comes in an octave away.

203
00:21:09,480 --> 00:21:11,730
Now I don't even want to says, I know you started with.

204
00:21:11,730 --> 00:21:17,280
So again, you just kind of set the pattern happening in the canon, so almost even has a spiral shape.

205
00:21:17,640 --> 00:21:21,290
And so for me on hearing this, there's a embedded inside the structure.

206
00:21:21,310 --> 00:21:28,080
This this is what we call a chorus of circles where the circles inside the structure of this piece and again,

207
00:21:28,080 --> 00:21:33,180
you know, the rhythms ball is very careful to to get all the different rhythms that are possible.

208
00:21:33,180 --> 00:21:35,129
So in the in the canons,

209
00:21:35,130 --> 00:21:44,040
the variations are broken off sometimes reasonable or reasonable or do these in them are the beats and divided into quavers, triplets or quavers?

210
00:21:44,340 --> 00:21:48,510
And what about going through the symmetries that if the two triangles are combination?

211
00:21:49,170 --> 00:21:58,260
He makes sure that each hand covers one of the possibilities either three B's and of all divide it up into seven quavers and Bach,

212
00:21:58,260 --> 00:22:03,570
the true imputation. Make sure that each and every different possibilities he is covered in brackets.

213
00:22:04,350 --> 00:22:09,810
Ball student medicine used to call box music a process, a sound in mathematics.

214
00:22:09,930 --> 00:22:15,330
And Bach isn't the only one who has lots of ideas symmetry in order to create variations.

215
00:22:15,510 --> 00:22:20,900
Very often the variations will mirror what is going on before when you were trying to push.

216
00:22:20,910 --> 00:22:26,129
And so symmetry is very often something you can hear embedded inside the music of, for example,

217
00:22:26,130 --> 00:22:31,830
Bach and another of the also I think, which uses symmetry or lost its architecture.

218
00:22:32,970 --> 00:22:38,870
We're in a building which really mixes art and mathematics here, police officers.

219
00:22:39,390 --> 00:22:43,050
We're right back to, in ancient times, the pyramids. In Egypt, for example.

220
00:22:43,230 --> 00:22:50,830
I looked we like these all Hadrian's Wall one hall stuck inside of the fence and he squares right through to modern day Paris.

221
00:22:50,980 --> 00:22:55,430
I think Paris is one of my favourite cities full of symmetry in fact.

222
00:22:55,430 --> 00:23:04,010
So the revolutionaries of the time of Galba had wanted to build a sphere at a hall to Paris because they felt that a sphere was

223
00:23:04,020 --> 00:23:13,410
a shape which represented the idea of égalité no direction favoured and over any other sort of perfectly symmetrical building.

224
00:23:13,830 --> 00:23:20,910
But since quite difficult to build architecturally on this one, eventually you'll be able to school Agios in Metz in Paris.

225
00:23:21,690 --> 00:23:29,350
I think the revolutionaries of debate just wanted to point out that this is in fact an all maximal revolutionary idea.

226
00:23:29,370 --> 00:23:37,620
But if you go to Israel, you actually find amazing blocks of flats which are based on the dodecahedron shape.

227
00:23:37,870 --> 00:23:41,759
The place you associate with the shape universe and solar panels is in Iran,

228
00:23:41,760 --> 00:23:48,210
not in Israel, although I'm told that these are hopeless furniture that I see.

229
00:23:48,600 --> 00:23:52,230
I went very closely with a mathematician in Japan called Professor Corncob,

230
00:23:52,470 --> 00:23:57,870
and I went out to all my studies and symmetry to go and do this work with him.

231
00:23:58,050 --> 00:24:04,470
And Professor Kurokawa took me up to his beautiful city, which I don't know whether any of you need to measure,

232
00:24:05,100 --> 00:24:09,660
which is full of all these beautiful Shinto shrines and Buddhist temples,

233
00:24:10,110 --> 00:24:14,310
and they're full of symmetry all over the place, the patterns inside the building.

234
00:24:14,490 --> 00:24:16,110
And as you all know, we took this photograph.

235
00:24:16,560 --> 00:24:23,500
We we climb the stairs to this archway and the Australia had these beautiful columns in these symmetrical designs.

236
00:24:24,390 --> 00:24:29,370
So eight columns exactly the same, except for one, which is turned upside down.

237
00:24:29,850 --> 00:24:34,190
And I said to Professor Kurokawa, Oh my gosh, that the architect must be very angry.

238
00:24:34,210 --> 00:24:37,830
The builders, I mean, go that one wrong. It's upside down. And this is not it.

239
00:24:37,860 --> 00:24:45,000
It was a very deliberate decision because in Japanese they love setting all these expectations and then embracing.

240
00:24:45,270 --> 00:24:50,309
And he referred me to this lovely book called The Japanese Essays in Idleness.

241
00:24:50,310 --> 00:24:53,670
What a wonderful title for a book from the 14th century.

242
00:24:54,150 --> 00:25:01,110
And he revealed this particular requirement on these. And then one thing uniformity is undesirable, leaving something incomplete.

243
00:25:01,390 --> 00:25:04,600
Interesting, given the feeling there is room for growth.

244
00:25:05,130 --> 00:25:09,610
Even when building the Imperial Palace, they always leave one place unfinished.

245
00:25:10,240 --> 00:25:15,310
Nancy even if you go back to the ball. Ball. GOLDBERG Variations seem to be full of symmetry.

246
00:25:15,640 --> 00:25:20,140
You reach the third variation. You sort of think you know exactly what's going to happen next.

247
00:25:20,470 --> 00:25:24,040
And then back to exactly the same thing turns the whole thing upside down like that.

248
00:25:25,210 --> 00:25:31,430
It's going to call it a musical check that this variation has absolutely nothing to do with the rest of the structure, too.

249
00:25:31,660 --> 00:25:37,209
And it's such surprise because you write it and then suddenly realise how much symmetry

250
00:25:37,210 --> 00:25:42,460
there being on that point that you appreciate that break in symmetry involved in that.

251
00:25:43,700 --> 00:25:49,880
But I think actually if I was going to be cast out, I had to leave the rest of my life in one building across the well.

252
00:25:49,910 --> 00:25:59,660
Being somebody who studies symmetry and love symmetry, I probably would choose the Alhambra in Granada because, you know, Hungary was not a palace.

253
00:25:59,660 --> 00:26:04,700
There was really a past century symmetry, this symmetry all over the place.

254
00:26:05,150 --> 00:26:10,640
There's lots of water throwing up reflections of the buildings in the water.

255
00:26:10,760 --> 00:26:22,280
They came to realise how, how, how special symmetry is because it was a little breath of wind for that symmetry suddenly to be shattered.

256
00:26:22,490 --> 00:26:29,000
So it's something very special. But I think it's funny, the walls where I really see symmetry being celebrated.

257
00:26:29,000 --> 00:26:33,470
So because all of these beautiful symmetrical tiles they use across the walls

258
00:26:33,890 --> 00:26:38,690
here in the offices exploring all the different possibilities that they have, think of the two candles.

259
00:26:39,110 --> 00:26:44,810
And actually, I think you see in the Alhambra, which is really where you can ask the really big questions about symmetry.

260
00:26:45,200 --> 00:26:49,100
Reporter What is a tree? What water?

261
00:26:49,310 --> 00:26:56,090
What does it mean to say that something is symmetrical? How can you say two things have the same symmetry.

262
00:26:56,690 --> 00:27:01,670
How do you understand if you actually around all the different symmetries that are possible?

263
00:27:02,600 --> 00:27:11,060
And as you go on to body language, you want more symmetry is is really what what Galileo's young revolutionary French mathematician did.

264
00:27:11,780 --> 00:27:18,670
I think there's a sort of feel that symmetry is all about that fine, rational symmetry, but there is so much more to this language.

265
00:27:18,680 --> 00:27:26,389
And it's what the guy, along with the one who was able to articulate where the symmetry and actually how I like to describe it is that

266
00:27:26,390 --> 00:27:35,300
symmetry is kind of something that you can do to an object such that when you finish doing something that will,

267
00:27:35,520 --> 00:27:40,850
it looks kind of like the tape we started with in a way, I kind of call it the magic trick.

268
00:27:41,090 --> 00:27:47,150
So if I give you, for example, if I show you this tetrahedron, what are the symmetries of this tetrahedron?

269
00:27:48,080 --> 00:27:51,500
Well, there's some moves that I can make to this tetrahedron.

270
00:27:52,160 --> 00:27:55,550
Sasha, when I made the move, it looks like it did before I saw it.

271
00:27:55,580 --> 00:27:59,660
So, for example, I could pick it up and I could do that.

272
00:28:00,710 --> 00:28:04,610
Now it looks like it did before I saw it. So that is a symmetrical move.

273
00:28:04,820 --> 00:28:10,280
So I like school symmetry, but I mean, I and you show your eyes, which are your eyes?

274
00:28:10,790 --> 00:28:17,700
Now I'm going to do something to this and open them. Now the tetrahedron looks like it did before you shut your eyes.

275
00:28:17,720 --> 00:28:22,120
But I did something to that which has this is one of these symmetries.

276
00:28:22,130 --> 00:28:27,980
So that's what I want you to think is symmetry is something you do to an object which brings it back to look like it did before.

277
00:28:27,980 --> 00:28:35,840
But actually it has changed something. So let's go to the walls in the Alhambra and try to understand these what symmetry really means.

278
00:28:36,350 --> 00:28:42,890
So, for example, this wall here, where's the symmetry here? Well, how we picked up the tiles and put them back down here.

279
00:28:43,370 --> 00:28:52,010
So if they look like for I move them. So I fix the tiles at this red point, pick them up and turn them by 90 degrees, pull them back down again.

280
00:28:52,400 --> 00:28:56,810
The tiles look like they were before I move them and that is a symmetry.

281
00:28:57,110 --> 00:29:05,600
So symmetry or all the structure is something that I can do to the objects, which means in some way makes it look like it did before I moved it.

282
00:29:06,350 --> 00:29:09,520
So I can take two very simple objects and we can do a small.

283
00:29:09,530 --> 00:29:13,730
So what are the symmetries that are made of these tools?

284
00:29:13,880 --> 00:29:17,300
So I'm going to take the object, a triangle and a 6.2.

285
00:29:17,300 --> 00:29:21,520
So I put a little twist off. So let's start with the six point itself.

286
00:29:21,560 --> 00:29:27,330
So for me, symmetry things. I can do movies, but it looks like you did before I knew this.

287
00:29:27,340 --> 00:29:31,070
So I put it together so I can keep track of what we're doing.

288
00:29:31,280 --> 00:29:35,020
So I can take the ten, put it back down again. And it looks like it did before we saw it.

289
00:29:35,030 --> 00:29:41,620
So that's one symmetry. Okay, well, I can move it by a third so I can take it around to see.

290
00:29:41,630 --> 00:29:51,110
So that's two symmetries. I can rotate it by half for 10 seconds, around two days as a third symmetry, I can do it by two thirds a turn.

291
00:29:54,030 --> 00:30:00,089
As for cemeteries and finally I can tell you do away rounds to five six to the fence of

292
00:30:00,090 --> 00:30:06,360
a565 moves that I'm going to make to this object such that it fits back down in size.

293
00:30:06,990 --> 00:30:10,979
Now there is a cemetery. Does anyone have an idea of six cemetery?

294
00:30:10,980 --> 00:30:15,600
I miss that. Yes. Second around the back to it.

295
00:30:15,870 --> 00:30:19,499
Yes, exactly. In fact, that's just leaving it where it is, isn't it?

296
00:30:19,500 --> 00:30:22,850
Because I could take it all the way round and then it hasn't moved at all.

297
00:30:22,860 --> 00:30:27,780
It's not so. So the same symmetry, he's actually just picking it up and put it back down here.

298
00:30:28,320 --> 00:30:33,210
Okay. You might say that's awesome, but actually it's a bit like zero zero.

299
00:30:33,510 --> 00:30:37,889
It's a very difficult concept as a number for mathematicians to get hold of because what's it counting?

300
00:30:37,890 --> 00:30:43,350
It isn't counting anything. So I want you to think of this as a like zero six, but everything has a symmetry.

301
00:30:43,350 --> 00:30:45,660
So zero is you just leave it all in.

302
00:30:45,670 --> 00:30:51,540
Even if you go to this of the irregular face, you see some symmetry which is just leaving the face levels, which is good.

303
00:30:51,750 --> 00:30:55,049
So I we say there are six symmetries there is going to break through for quite a

304
00:30:55,050 --> 00:31:00,030
while to think of just leaving it as a six symmetry and you can't pick the seven,

305
00:31:00,150 --> 00:31:02,840
for example. That might have been one of your other thoughts that it has.

306
00:31:03,090 --> 00:31:07,799
It hasn't got a reflection of symmetry, because if I flipped over, it's pointing the other way.

307
00:31:07,800 --> 00:31:12,630
So it hasn't got another symmetry coming from reflection, however, I won't take the triangle instead.

308
00:31:12,840 --> 00:31:17,760
This does have what is called two rotation symmetries. I'm going to rotate my third clockwise.

309
00:31:17,950 --> 00:31:20,190
I want to rotate my third anticlockwise,

310
00:31:20,370 --> 00:31:25,469
but now I've got refraction so I can take the triangle flipped over and I'm going to go three ways I can do that.

311
00:31:25,470 --> 00:31:30,330
I can flip it over through the line, through X or get it over through the line through y.

312
00:31:30,600 --> 00:31:34,320
So the X is in point swap or I could fix it and put it over there.

313
00:31:34,620 --> 00:31:42,149
So that's five different symmetries. Again, two rotations for three reflections and then I six symmetry, which is just leave me where it is.

314
00:31:42,150 --> 00:31:51,299
I pick it up, put it back down again. So it's got six entries. So this is quite interesting because these two objects both have six symmetry.

315
00:31:51,300 --> 00:31:56,490
So perhaps we should say they have the same symmetry because they have the same amount of symmetry.

316
00:31:57,450 --> 00:32:02,730
But what I want is to explore why actually these two how we articulate really that no,

317
00:32:02,790 --> 00:32:06,210
these, these objects are two genuinely different symmetrical objects.

318
00:32:06,450 --> 00:32:11,130
But before I did I say a little challenge, which I want you to think about through the rest of the lecture,

319
00:32:11,140 --> 00:32:16,270
not, not see that this is the other stuff, but now you've got some idea of magic for me.

320
00:32:16,360 --> 00:32:19,530
So I want you to, to consider another object.

321
00:32:19,560 --> 00:32:22,650
I want you to consider the Rubik's Cube. Rubik's Cube.

322
00:32:22,650 --> 00:32:26,730
So what does symmetries of the Rubik's Cube? Well, think about this magic trick.

323
00:32:26,880 --> 00:32:30,690
So no colours. I don't care, of course, but a symmetry in the cube.

324
00:32:30,690 --> 00:32:35,280
It's really just anything I can do to the cube, which makes it look like still cube.

325
00:32:35,640 --> 00:32:38,910
So I could do that fancy symmetry. Or I could do that.

326
00:32:38,910 --> 00:32:42,190
That's. That's. And that's enough.

327
00:32:42,970 --> 00:32:46,960
So what I want you to think about, there'll be a prize for the person who gets places.

328
00:32:47,200 --> 00:32:53,740
How many symmetries do you think this thing has? How many different, genuinely different moves could I make to this cue?

329
00:32:54,370 --> 00:32:57,819
Such it still looks like a cue. We'll come back to that later.

330
00:32:57,820 --> 00:33:07,270
And I've got a price with this in this case. But so let's come back to these two shapes, because the real breakthrough got along.

331
00:33:07,630 --> 00:33:16,720
First of all, to think of of of symmetry is something about motion as you went very much against what Thomas Mann but for Thomas Mann,

332
00:33:16,720 --> 00:33:20,690
symmetry was something deathly something which didn't move but actually got lost.

333
00:33:20,690 --> 00:33:24,190
And now you should think about symmetry as something with lots of movements in it.

334
00:33:24,250 --> 00:33:30,790
It's all about it's all about life. It's all about the things that you can do as an object, which it looks like it did before I knew it.

335
00:33:31,480 --> 00:33:36,790
But the second great breakthrough was that he realised it isn't just the individual symmetries that are important,

336
00:33:37,480 --> 00:33:42,020
but how they interact with each other. So what do I mean by that?

337
00:33:42,050 --> 00:33:47,600
Well, if you think about it, if I do a magic trick, move to shape, then I can do another move to it.

338
00:33:47,870 --> 00:33:51,650
And then the combined effect is actually a third magic trick.

339
00:33:51,920 --> 00:33:56,360
So actually, all of these things are related. If I do one move followed by another.

340
00:33:56,570 --> 00:34:00,110
The combined effect is is divided in a third symmetrical move.

341
00:34:00,500 --> 00:34:07,610
So in fact, Galois began to explore, okay, what was the relationship between the 66 symmetries of the sixth point you saw?

342
00:34:07,800 --> 00:34:12,180
What's the relationship between all of those symmetries? If I do one followed by another.

343
00:34:12,200 --> 00:34:15,650
It's got to be a third symmetry, so it must be one on that list.

344
00:34:15,980 --> 00:34:18,500
So I can two names to all of these symmetrical moves.

345
00:34:19,010 --> 00:34:28,280
So the name Amy goes to just leaving the thing where it is correct to be moving it by a sixth, but to see a third of a and so on.

346
00:34:28,880 --> 00:34:34,070
So now let's see what happens if I combine a six of inside followed by ten.

347
00:34:34,460 --> 00:34:40,710
So I'm going to do I'm going to do count will be first is equal to ten and I'm going to do capital C a third.

348
00:34:41,240 --> 00:34:47,210
So we do a combined effect and this is what this table records, sixth of a term followed by a third of a time.

349
00:34:47,450 --> 00:34:50,690
Well, the combined effect is actually about the same half term in one go.

350
00:34:51,500 --> 00:34:55,760
So actually, this kind of algebra, a language hiding behind these symmetries,

351
00:34:55,760 --> 00:34:59,600
which says, if I can combine two sentences, it's going to be a third symmetry.

352
00:34:59,600 --> 00:35:02,910
And this table records for you how these symmetries interact.

353
00:35:02,930 --> 00:35:07,790
So the combination of being followed by C is actually the same as just in the half step.

354
00:35:08,300 --> 00:35:11,430
Now I want to look do it in the other order with a three mass.

355
00:35:11,450 --> 00:35:14,509
I could have done the third return first, followed by the same for the turn.

356
00:35:14,510 --> 00:35:17,930
And still I get half as it really doesn't matter. What do you do?

357
00:35:18,230 --> 00:35:23,510
And in fact, you can see that reflected in this table. The table records all of the different pairs of symmetries.

358
00:35:23,510 --> 00:35:27,410
And what the effect is, is the symmetry. Whatever you. Doesn't matter what order you do.

359
00:35:28,640 --> 00:35:31,820
However, if we come to the triangle, it does matter. What will you do?

360
00:35:32,060 --> 00:35:39,110
So I'm going to do a third button clockwise that's going to be you go and you follow by flip and the reflection in X,

361
00:35:39,260 --> 00:35:42,980
the combined effect is actually as if the triangle was relaxing it.

362
00:35:43,310 --> 00:35:49,880
But if we do it in a different order, let's do the reflection in experts and then the rotation by third anticlockwise.

363
00:35:50,690 --> 00:35:55,070
Actually, that's the triangles ended up in a different place. It's a table like reflected in Y.

364
00:35:56,090 --> 00:36:01,130
If I. If it is if it goes to the part later, I want you to take a b-max and look at the picture on the bottom.

365
00:36:01,310 --> 00:36:06,670
And then what I want you to do is do a 90 degree turn clockwise and then a reflection in a line away from you,

366
00:36:06,780 --> 00:36:12,829
and then see when the picture is pointing, put it back, back to where it was, and then do the reflection first and then a rotation.

367
00:36:12,830 --> 00:36:18,770
You find that the picture is pointing in the other direction. So here it matters what order you do the symmetries in.

368
00:36:19,100 --> 00:36:26,120
And so it's the way the symmetries interact that we're able to start to distinguish why the symmetries of the triangle that genuinely difference,

369
00:36:26,130 --> 00:36:33,020
the symmetries, all the six points are. In fact, what I want about was a set of rules for the way the symmetries interact.

370
00:36:33,320 --> 00:36:37,130
And he shows that it's an object with six symmetries.

371
00:36:37,340 --> 00:36:42,950
Either the six symmetry must behave like the 6.2 starfish, or they must behave like the triangle.

372
00:36:43,190 --> 00:36:46,620
And there isn't a third symmetrical object. It will either.

373
00:36:46,620 --> 00:36:50,600
The symmetry will interact in the same way as one of these two objects.

374
00:36:51,350 --> 00:36:57,680
So suddenly this language now you've enabled him to say, Yeah, actually there are only two symmetrical jets with six symmetries.

375
00:36:58,100 --> 00:37:02,389
If I could use this language, it's a bit like a set of rules for Sudoku, for example.

376
00:37:02,390 --> 00:37:09,560
Let me do it. Inside the body of this table, you never see a symmetry twice in a line or in a corner.

377
00:37:09,980 --> 00:37:13,910
And so Galileo developed some set of rules to the way symmetries must interact,

378
00:37:14,090 --> 00:37:18,350
which helps him to then explore, for example, symmetries in the Alhambra.

379
00:37:18,590 --> 00:37:22,340
So it took actually till the end of the 19th century for us to use Galois language.

380
00:37:22,730 --> 00:37:26,900
But for example, these tools here, these two always look very different.

381
00:37:27,230 --> 00:37:33,170
One is made up of these kind of shimmering triangles, the other six point to solve these shapes.

382
00:37:33,380 --> 00:37:40,430
But actually using Galileo's language, we can say that these two tilings actually represent the same group of symmetries.

383
00:37:40,880 --> 00:37:44,450
So where all the symmetry decided, let's take a guess and measure moves.

384
00:37:44,750 --> 00:37:48,950
So there's one point where I can rotate by six and ignore the colours.

385
00:37:49,130 --> 00:37:53,150
So think of a turn to the point where all the triangles meet. That's one symmetry.

386
00:37:53,330 --> 00:37:58,540
There's another symmetrical move I can make, which is to take the points in the middle of the triangles, rotate one third of the time,

387
00:37:58,730 --> 00:38:06,410
the sun will look down on top of each other, and then an isolated corner place is fine, but halfway along the edge you can flip these by half a turn.

388
00:38:06,620 --> 00:38:11,719
And so what happens now is that the triangles sort of swap places on top of each other.

389
00:38:11,720 --> 00:38:15,950
So the white triangle went to the blue and blue went and said once and that's a third move.

390
00:38:15,950 --> 00:38:21,020
So basically these are the ingredients and all the symmetries of this particular set of toys.

391
00:38:21,350 --> 00:38:25,550
But if we go to the other wall, which I showed you, well, the same moves work here.

392
00:38:25,730 --> 00:38:29,720
There's a six out of a ten you can make at the centre of a six pointed star.

393
00:38:30,350 --> 00:38:35,030
There's a fair return. If you look where the pieces meet, you can rotate by a third return there.

394
00:38:36,020 --> 00:38:40,130
So that's point there. And then again, after you take the halfway point between six.

395
00:38:40,250 --> 00:38:43,850
Point in stores. That's another place where you can make these rotations.

396
00:38:44,150 --> 00:38:49,040
So although these pictures look completely different, actually using our language,

397
00:38:49,040 --> 00:38:53,719
we can articulate the symmetries of these two walls or exactly the same facts.

398
00:38:53,720 --> 00:39:01,070
In this case, we can put one tiling on top of another, and any move that we made to one will actually represent the symmetry of the other.

399
00:39:01,730 --> 00:39:02,390
So these two,

400
00:39:02,630 --> 00:39:09,620
I think you're going to think of Gauloises work a little bit like the concept of number and the concept of number is quite an abstract idea.

401
00:39:10,040 --> 00:39:14,180
For example, level three people in the front row here sitting in three chairs.

402
00:39:14,510 --> 00:39:20,210
Now the people of right difference the chairs, but the abstract idea of the number three is common to both.

403
00:39:20,960 --> 00:39:26,480
So symmetry, I want you to think of it in a similar way. It's not about geometry in pictures.

404
00:39:26,630 --> 00:39:30,380
These two pictures that I showed you, they have the same underlying symmetries,

405
00:39:30,380 --> 00:39:36,470
but they're different representations of those symmetries just in the same way as we got different representations of the number three.

406
00:39:36,910 --> 00:39:41,360
So it enables us to take, for example, these three different images.

407
00:39:41,360 --> 00:39:46,430
There's one is the tilings, the entrance to the Alhambra, one's the floor, one's the ceiling.

408
00:39:47,120 --> 00:39:52,700
Actually, these will have the same symmetrical moves that can be made, all of them, although the pictures look very different.

409
00:39:53,000 --> 00:39:59,690
This is a symmetry for 14. It's nothing to do with football but to do with the fact that there's a place where you can do a quarter of a turn.

410
00:39:59,810 --> 00:40:02,100
Another place? A quarter of a ten and a half of ten.

411
00:40:02,990 --> 00:40:09,110
So this language suddenly gave us a way to articulate when two different things might look very different,

412
00:40:09,110 --> 00:40:17,180
actually have the same underlying symmetries. By the end of the 19th century and allowed us to actually articulate the fact that in the Alhambra,

413
00:40:17,600 --> 00:40:24,650
there are only 17 different sorts of underlying sentences that you can make on a two dimensional wall.

414
00:40:25,400 --> 00:40:27,410
And if you think about those pictures of Escher,

415
00:40:27,530 --> 00:40:34,489
as he was very inspired by his trips to the Alhambra and he made lots of different pitches with devils and angels,

416
00:40:34,490 --> 00:40:43,010
a little selfish and efficient garden. But actually we can prove mathematically that however, on Escher drawings or anyone else,

417
00:40:43,010 --> 00:40:47,330
they will never be able to find anything more than the 17 that are on this list.

418
00:40:47,690 --> 00:40:53,710
And interestingly, what I do these in community holidays is my family, which they hate.

419
00:40:53,720 --> 00:40:55,730
But I took them to the Alhambra and I said, okay,

420
00:40:55,760 --> 00:41:01,489
we can see where they actually found 17 different symmetries in the Alhambra and they almost stayed there.

421
00:41:01,490 --> 00:41:06,830
The last one is Sol three, three, three, and those blue tiles, we need to paint those black.

422
00:41:06,950 --> 00:41:11,770
And then we would have had an indication that we didn't actually see this abstractly.

423
00:41:11,780 --> 00:41:18,439
But but it's amazing. That's actually even the artist was kind of ahead of the mathematicians in being able to find all of these.

424
00:41:18,440 --> 00:41:24,320
But it was only with mathematics that we're able to articulate the sort of limits of this world of symmetry.

425
00:41:25,670 --> 00:41:33,220
The only thing Gabor did was to actually realise that symmetry also has a kind of atomic symmetries,

426
00:41:33,240 --> 00:41:37,040
a bit like the possibility the periodic table of symmetry.

427
00:41:37,250 --> 00:41:41,400
Now, I think it's actually extraordinary and I studied Galileo to realise this.

428
00:41:42,210 --> 00:41:50,360
This man, aged 20, had come up with so many amazing ideas and this is perhaps the most savvy one because he showed why a

429
00:41:50,360 --> 00:41:56,810
symmetrical object can actually be broken down into the symmetries of smaller atomic symmetrical objects.

430
00:41:57,620 --> 00:42:02,720
And this gave us a way in somehow really get to grips with the world of symmetry mathematically.

431
00:42:03,020 --> 00:42:05,900
So, for example, here's a 15 sided coin.

432
00:42:06,200 --> 00:42:13,520
The symmetries in that 15 sided point, you now know how to do their sort of rotations of a 15 sum to four things and things like that.

433
00:42:13,790 --> 00:42:23,210
But actually, you can make the symmetry of this 15 sided coin at the centrepiece of two options, namely a pendulum and applying so and so on.

434
00:42:23,690 --> 00:42:30,800
So actually symmetries of the paintings. I do think that I can do all of them by doing Symmetries of the Pencil, a triangle.

435
00:42:31,040 --> 00:42:36,290
For example, I want to rotate the Pentagon by 50 return.

436
00:42:36,290 --> 00:42:41,170
So the key here is the fact that 15 can be broken down into the Primes three and five.

437
00:42:41,660 --> 00:42:48,920
So let's say I want to rotate by 5610. So I want to get let's say we're going to get six green dots.

438
00:42:48,920 --> 00:42:53,580
515 is the ten. I want to get it to yellow. I'm going to use the symmetry, symmetries, the triangle, the principle.

439
00:42:53,930 --> 00:42:57,980
So I'm going to take over the pendulum. I'm going to rotate by 2/5 to the 10th.

440
00:42:58,280 --> 00:43:00,710
So my brain is going to go down to the blue dot,

441
00:43:01,070 --> 00:43:06,860
but now I'm going to let the triangle take over and I'm going to do a rotation, my third return and equalise.

442
00:43:07,220 --> 00:43:15,860
And then also the combination of those two, hey presto, I've got a 15 for this year and this basically is at the half of all of the symmetry.

443
00:43:16,070 --> 00:43:18,139
You say any symmetrical object, actually,

444
00:43:18,140 --> 00:43:25,210
you'll be able to find smaller symmetrical objects inside of it whose symmetries can be used to build out the symmetries of the large objects.

445
00:43:25,670 --> 00:43:31,790
If you just like two dimensional flat shapes like this, basically it's the primes which are the atomic shapes.

446
00:43:32,240 --> 00:43:39,020
So any shape of 105 sides, you can make that out of triangle Pentagon 675 theta.

447
00:43:39,650 --> 00:43:47,720
So it may. Daniel's discovery that he could do this gave us the idea that you could produce a periodic table of symmetry.

448
00:43:47,870 --> 00:43:53,510
Now we can try to classify what all of these building blocks are symmetry on to understand the whole of the symmetrical world.

449
00:43:54,470 --> 00:43:59,390
So. So already we found the primes of their prime sided shapes of some of the building blocks.

450
00:43:59,780 --> 00:44:05,630
It turns out they're not the only ones. In fact, there are some much more exotic, little more complicated ones.

451
00:44:05,930 --> 00:44:16,670
And intriguingly, the first sort of interesting prime shape in all this is the symmetries of the icosahedron.

452
00:44:16,970 --> 00:44:20,410
Well, actually, not to say that usually the only considering you need to put a slight twist on it.

453
00:44:20,420 --> 00:44:28,820
So this is actually a chocolate box that Escher designed for a manufacturer in Holland of celebrating an anniversary.

454
00:44:29,390 --> 00:44:35,660
When I retire, I want one of these boxes asking you to go to the SWC and Den Haag.

455
00:44:35,840 --> 00:44:43,010
They've got an example, I presume, that makes many of these and the little tin boxes, but they're the most beautiful things.

456
00:44:43,010 --> 00:44:50,540
And but the symmetries of this box are actually one of the first sort of non obvious point symmetries that you can have,

457
00:44:50,720 --> 00:44:58,610
because actually, how many symmetries is this object have? Well, actually, it's the same as the centuries old football.

458
00:44:59,060 --> 00:45:04,420
So the football his football made out of pentagons and and hexagons.

459
00:45:04,430 --> 00:45:09,410
But actually, the symmetries of this are the same as the symmetries of the icosahedron.

460
00:45:09,420 --> 00:45:12,499
So how many different moons on that? I'll call with Max Lucas.

461
00:45:12,500 --> 00:45:18,060
Exercise. Quite a twist. So he's twisted this starfish, so there isn't the reflection, like my 620 starfish.

462
00:45:18,470 --> 00:45:22,430
So how many different moves on there that I can make? You know, I can I can do that.

463
00:45:23,360 --> 00:45:31,640
So I'm just looking to read things all like you find it when you do this, you can do 60 different moons, 60 different symmetries of this shape.

464
00:45:32,410 --> 00:45:35,480
Now, come on, 60. That's an incredibly divisible number.

465
00:45:36,830 --> 00:45:40,410
The reason the Babylonians use it for the base of the nervous system, because it's so obvious.

466
00:45:40,520 --> 00:45:49,040
It's the reason we have 60 minutes in the air. So but when you consider the symmetries of this object, it turns out that number 60 is very invisible.

467
00:45:49,310 --> 00:45:54,230
Actually, the symmetries of this can't be divided into smaller symmetrical objects.

468
00:45:54,470 --> 00:45:58,380
You might say, well, hold on. I mean, there's a painting in here.

469
00:45:58,400 --> 00:46:05,180
Surely the symmetries of the Pentagon. I just see that that's a kind of symmetry, which is a smaller hiding inside.

470
00:46:05,810 --> 00:46:10,880
But if I tried to divide by the Pentagon and trying to symmetrical objects, it doesn't make any sense.

471
00:46:10,910 --> 00:46:17,060
Turns out the mathematics that go on to balance that there's no way to break this down into smaller objects.

472
00:46:17,210 --> 00:46:23,060
This turns out to be one of the first kind of interesting atoms in the periodic table of symmetry.

473
00:46:23,390 --> 00:46:29,540
Now, it's got a lot of studies on this one, but for 150 years, it's got a long way to go.

474
00:46:29,840 --> 00:46:35,870
We were trying to classify more and more these natural objects, trying to find the acids which make up symmetry.

475
00:46:36,020 --> 00:46:45,169
And it culminated in the 1980s with what we believe was the completion of operating a table of US extraordinary objects,

476
00:46:45,170 --> 00:46:56,060
including this amazing object, which is a and I'll show it to you because it's an object is a symmetrical object, which is a 996,893 dimensional size.

477
00:46:56,690 --> 00:47:02,530
Even in that sense, you can call it altered for you physically, but amazingly, isn't it?

478
00:47:02,930 --> 00:47:09,530
If you want to think about is this kind of weird movement which exists in an unusual size and symmetries.

479
00:47:10,010 --> 00:47:13,339
So the more symmetries that they are acting as the sun gets,

480
00:47:13,340 --> 00:47:21,260
we have to suppose we couldn't write this object and just lose interest and we just didn't have any pauses,

481
00:47:21,260 --> 00:47:25,010
just weird objects that suddenly appeared at this point in space.

482
00:47:26,780 --> 00:47:34,690
But how does the mathematicians say that's what I mean by 196,883 objects?

483
00:47:34,700 --> 00:47:38,270
I mean, how do mathematicians play around with objects and stuff point on it?

484
00:47:39,470 --> 00:47:45,380
Well, I think this is another example of how fantastic language is in mathematics and language.

485
00:47:45,590 --> 00:47:50,400
In a way, you have got lost the physical world symmetry of algebra.

486
00:47:50,780 --> 00:47:58,929
Suddenly the pictures disappear. I have access to the table which allows us to sort of see mathematically shapes.

487
00:47:58,930 --> 00:48:05,600
And I mentioned school and school of easy geometry, Cartesian geometry space to be a wonderful dictionary,

488
00:48:05,810 --> 00:48:10,040
a language which translates a lot of shapes into the world of numbers.

489
00:48:10,730 --> 00:48:20,450
So, for example, if I wants to describe a square in numbers, I can locate a school in a sort of east, west, North-South south location.

490
00:48:21,410 --> 00:48:34,489
So the square consists of a point at the origin 001 set along the horizontal axis, 101701 and that is it.

491
00:48:34,490 --> 00:48:39,950
One line where I find something that. So in a way I can translate a square into the.

492
00:48:40,590 --> 00:48:44,970
Seeing results pairs among school passengers identify for you this way.

493
00:48:45,930 --> 00:48:53,660
If I want to go off the Dimension Cube where I can draw you key, but I can similarly use the dictionary to translate it into numbers of Americans.

494
00:48:53,910 --> 00:48:59,550
AIDS triples in numbers. Which car? As long as you know how many moves you make in each direction.

495
00:48:59,790 --> 00:49:03,719
No, no, no. One, nought, nought, nought, nought, nought, one.

496
00:49:03,720 --> 00:49:08,160
All the way through to the stream point while online. Okay.

497
00:49:08,280 --> 00:49:09,750
What about a four dimensional cube?

498
00:49:10,170 --> 00:49:17,630
Now, the point is, this dictionary is wonderfully powerful, because although the visual side runs out of the three, I don't show you all of it.

499
00:49:18,300 --> 00:49:21,510
Still, the numerical by the numbers carries on.

500
00:49:21,540 --> 00:49:32,690
I can tell you in numbers, one four dimensional cube is it's a it's a shape which has points light occasions 0000.

501
00:49:33,150 --> 00:49:43,230
And then from that point, you got an extra now two points at 1000010000100001.

502
00:49:43,470 --> 00:49:50,520
So for edges that I just keep on going. Maybe six in each of these directions it's going at some point at one, one, one, one.

503
00:49:51,280 --> 00:49:54,900
Now, how many corners has a four dimensional cubicles?

504
00:49:56,010 --> 00:50:01,650
16 central. You able to work out how many different ways there are putting zeros and ones inside?

505
00:50:02,040 --> 00:50:06,240
16. You could also use the same trick to look at how many edges there are.

506
00:50:07,110 --> 00:50:13,800
Now you might have to explore very quickly using this language the geometry of a shape you cannot see.

507
00:50:15,070 --> 00:50:18,480
And little thought should be able to get around the edges that are all there.

508
00:50:18,780 --> 00:50:26,250
And that's the amazing thing that we can all see. Move to symmetries of this automation cube by using the same coordinates.

509
00:50:26,590 --> 00:50:29,890
So now I kind of have to show you a shadow of this shape.

510
00:50:29,910 --> 00:50:34,920
So, in fact, if you go back to Paris, there is a shadow of a four dimensional cube.

511
00:50:35,310 --> 00:50:39,090
Actually, the financial area in Paris.

512
00:50:39,510 --> 00:50:42,749
The all points is a shadow of information.

513
00:50:42,750 --> 00:50:46,980
Q In three dimensions. Do you think about a three dimensional cube?

514
00:50:47,280 --> 00:50:51,660
Actually, then all this is really three dimensional cube on a two dimensional capitals.

515
00:50:51,810 --> 00:50:57,389
What they do is make a square and they put a smaller square inside and then they don't see what it says.

516
00:50:57,390 --> 00:51:01,350
Because you think the same cube, because it's just a black thing.

517
00:51:01,690 --> 00:51:06,130
This is the same sort of concept a project organising cube has in three dimensions.

518
00:51:06,160 --> 00:51:10,590
What you get is a small view inside an object view and the size will join up.

519
00:51:10,960 --> 00:51:18,150
So I have an example here. Here's a four dimensional cube projected down into a three dimensional universe.

520
00:51:18,490 --> 00:51:24,060
I can see you correctly articulated that there are 16 corners and you can see them

521
00:51:24,240 --> 00:51:28,890
nice little whites balls and actually you'll be able to count all the edges.

522
00:51:29,190 --> 00:51:32,730
Now. So yeah, I do the maths.

523
00:51:33,120 --> 00:51:37,080
So the 12 inch cube on there.

524
00:51:37,470 --> 00:51:42,420
So that's 24, 26, 31, 32, 32 edges.

525
00:51:42,600 --> 00:51:48,010
But actually, you could have done that. Now, this picture, if I tried, was if I want an edge in terms of these.

526
00:51:49,350 --> 00:51:53,820
But it's going to be around 196,883 are actually shadows.

527
00:51:54,240 --> 00:51:57,790
And what's good for you. So I'm trying to make small symmetries of this shape.

528
00:51:57,810 --> 00:52:06,450
You really do have to use a mathematical language which translates geometry into numbers, and amazingly, you navigate out of it.

529
00:52:06,630 --> 00:52:13,560
Even measure is pretty an amazing pleased to be able to explore the symmetries and all of this number branches.

530
00:52:13,830 --> 00:52:17,219
And one of the most is of the will of symmetry.

531
00:52:17,220 --> 00:52:27,750
You have to complete this kind of classification over the centuries to incorporate the masses of all symmetry.

532
00:52:27,900 --> 00:52:33,060
In fact, when I finished my degree area in offices, I mean, I've fallen in love with world symmetry.

533
00:52:33,240 --> 00:52:40,470
It was now 1985, and I just heard about this sort of completion of this project.

534
00:52:41,250 --> 00:52:50,340
The Cambridge seemed to be the place to go to study symmetry, and in fact, they produced this amazing piece that has been around is the real thing.

535
00:52:50,940 --> 00:52:57,180
The Atlas five is this basically inside here are all the building blocks.

536
00:52:57,660 --> 00:53:04,270
This is the periodic table of symmetry was produced in about 1985, just when I was finishing my three.

537
00:53:04,310 --> 00:53:12,120
And I went up to Cambridge because John Conway and his group to try and find out, you know, what are the next steps after having done this.

538
00:53:13,290 --> 00:53:14,820
So I was very keen to join that group.

539
00:53:16,470 --> 00:53:25,440
And he sat down and he said, well, you know, we are all very obsessed with symmetry, so let's make out at the same time.

540
00:53:25,830 --> 00:53:39,879
He said, Oh. We have to drop the do something against Rangers, though, because I know the authors in this area have six letters on each word.

541
00:53:39,880 --> 00:53:47,530
I don't like two schools in the same place when I say, you know why, but nobody's going to do that.

542
00:53:48,040 --> 00:53:52,990
I was like going agree. So as bad as it is to and others.

543
00:53:53,680 --> 00:53:58,080
Yeah. And if you drop one of those letters.

544
00:53:58,520 --> 00:54:04,809
Walter, we only have two games on the air, so I'm not an MP.

545
00:54:04,810 --> 00:54:11,379
So my argument is, if that's not all, that's the first to join the group with John Conway.

546
00:54:11,380 --> 00:54:17,620
And then he got his Ph.D. student, Ralf Hedges, that it was this guy hanging around and Cambridge just kept running around.

547
00:54:17,850 --> 00:54:23,559
It seemed to be quite helpful. Simon Norton So he was the third all go joined.

548
00:54:23,560 --> 00:54:27,940
He was really good at computing. The things he wore them all through joined with Wilson.

549
00:54:28,450 --> 00:54:34,999
They all joined in alphabetical order. So if I was going to join the group, I was going to have to change my name off.

550
00:54:35,000 --> 00:54:44,760
The same for but since. So now about that point I realise well I know that's going a little bit too far though,

551
00:54:45,940 --> 00:54:53,999
and so I'm a little bit concerned that I came back to work since I wanted to start my research into the world of surgery.

552
00:54:54,000 --> 00:54:57,320
And I, I want to go maybe. Maybe surgery finished.

553
00:54:57,420 --> 00:55:03,420
Maybe we all understand everything there is to know about, but that turns out not to be true is never true.

554
00:55:04,140 --> 00:55:10,290
So one problem that opens up so many more interesting columns in particular, particularly single who monster.

555
00:55:10,670 --> 00:55:16,090
It was this strange thing, right? In 196,883 inches right.

556
00:55:16,170 --> 00:55:22,560
Here is why you suddenly suddenly emerge in this financial space and it has this number or any significance,

557
00:55:23,190 --> 00:55:28,740
but it turns out it does this number you find in a completely different area of mathematics called number theory.

558
00:55:28,980 --> 00:55:33,790
This number comes up in some modular forms, which is very much related to the work,

559
00:55:33,790 --> 00:55:40,470
what I wanted on Fermat's Last Theorem and even conflate the other dimensions with somebody, this monster appears.

560
00:55:40,710 --> 00:55:46,110
All the numbers of the dimensions have something to do with this completely different area of mathematics.

561
00:55:46,890 --> 00:55:51,750
And so one of the challenges has been we made some progress, but I still think there's a lot to unlock.

562
00:55:51,930 --> 00:55:57,749
Why has this strange metrological anything to do with the world of number theory in school?

563
00:55:57,750 --> 00:56:05,610
Monstrous moonshine. Because John Connolly thinks that there's somehow a third object which is shining lights on both of these things,

564
00:56:05,610 --> 00:56:09,300
which in the moonlight we see is these strange numbers appearing.

565
00:56:09,690 --> 00:56:17,370
In fact, Richard Fortune's Won the Fields Medal or Nobel Prize for making a lot of discoveries about what?

566
00:56:17,370 --> 00:56:20,729
That the sun might be shining a light on a clump.

567
00:56:20,730 --> 00:56:28,590
I still believe that there's a lot we still do not understand about this strange symmetrical object, which is sort of one of the strange things.

568
00:56:28,690 --> 00:56:35,790
This happens on my symptoms, but actually my own research I might be interested in.

569
00:56:36,120 --> 00:56:40,470
If this is the periodic table, then what are the molecules of symmetry that you can make?

570
00:56:40,650 --> 00:56:45,300
What can you build? What are the new sort of trees that you can make out of this list in the spirit table?

571
00:56:45,330 --> 00:56:50,280
So as you you know, I spend more time here analysis and doing research and symmetry,

572
00:56:50,280 --> 00:56:57,120
trying to build new symmetry out of the the building blocks that we got a in the act as a simple so and

573
00:56:57,120 --> 00:57:02,249
so I discovered a strange using that for objects with connections to things called related curves.

574
00:57:02,250 --> 00:57:08,400
And in fact I got one of these Newsmax.com objects which hasn't got a name yet,

575
00:57:08,430 --> 00:57:12,780
and this is going to be a price for you that I'm going to name the symmetrical object.

576
00:57:12,780 --> 00:57:19,140
Also the person who can work out how many symmetries that Rubik's Cube has.

577
00:57:19,440 --> 00:57:25,979
So I got a little certificate here. It's not a name yet, so that is something.

578
00:57:25,980 --> 00:57:30,010
Group. This could be your name. No, no. Modern science.

579
00:57:30,610 --> 00:57:35,669
Science is a kind of a process of evolution as that one theory gets overturned by another.

580
00:57:35,670 --> 00:57:41,760
You know, the science the ancient Greeks proved in law school of business.

581
00:57:42,090 --> 00:57:48,810
The amazing thing about mathematics is it's eternal. This group will be there forever and species will die.

582
00:57:48,840 --> 00:57:54,410
Stars will up. But this group may give you a little bit of immortality if you can claim some.

583
00:57:54,780 --> 00:57:58,320
So the challenge is, okay, how many symmetries does this Rubik's Cube have?

584
00:57:59,380 --> 00:58:03,780
Okay, so I've put it back there, the Rubik's Cube here now.

585
00:58:03,780 --> 00:58:05,990
So how does I'm going to do this? Okay.

586
00:58:06,240 --> 00:58:17,010
What I want you to do is if you go to my guess, I just want you to make a guess at how many digits this number has.

587
00:58:18,090 --> 00:58:27,389
So. So just if you count I did a number of women make a guess how many digits do you also think it's got?

588
00:58:27,390 --> 00:58:32,430
A hundred symmetry. So it's got three digits. Okay, now you want to play you please stand up.

589
00:58:32,760 --> 00:58:39,180
I'm going to try and sort it out. So make your estimates using all three symmetries or three in our series.

590
00:58:39,270 --> 00:58:45,390
Right. So we've got some sort of players or we need to guess and you will get to be able to see if you get close to the right.

591
00:58:45,840 --> 00:58:49,139
So you only have one estimate. I don't need a precise formula.

592
00:58:49,140 --> 00:58:52,470
I just want you to estimate how many digits you think there are.

593
00:58:52,950 --> 00:58:58,380
Okay, so I'm going to sort it out. I'm used to calculating the but very impressive.

594
00:58:59,670 --> 00:59:06,180
So if you go to ten or fewer digits, I want you to sit down.

595
00:59:06,420 --> 00:59:11,950
Underestimated the number. So more than that.

596
00:59:12,120 --> 00:59:20,940
So if you now if you got 30 or more digits, you also sit down because you overestimate.

597
00:59:21,240 --> 00:59:25,290
And so it's the numbers. It's between ten and 30.

598
00:59:26,070 --> 00:59:32,940
Okay. So if you've got some, uh, more than 20 digits in your number.

599
00:59:34,110 --> 00:59:38,060
I want you to keep standing. All right.

600
00:59:38,430 --> 00:59:41,880
So it's between. Right. So you go there. Sometimes you got to sit down.

601
00:59:42,160 --> 00:59:50,130
So. Okay. If you've got more than 28 tickets and you have a you want to sit down.

602
00:59:52,040 --> 00:59:55,630
27. 23.

603
00:59:55,670 --> 00:59:59,850
Sit down. Okay. Right. Right.

604
00:59:59,900 --> 01:00:02,900
So it's between. It's a it's more than 23. Okay.

605
01:00:03,160 --> 01:00:06,959
It's alright. So how many of you. Tony.

606
01:00:06,960 --> 01:00:10,870
So you got to sit down to sit down. 24.

607
01:00:11,780 --> 01:00:15,349
24, 24, 25, 25.

608
01:00:15,350 --> 01:00:19,310
This is great news. So what is your name, sir?

609
01:00:19,520 --> 01:00:23,060
It's actually has 26 inches 25. Exactly.

610
01:00:23,300 --> 01:00:26,690
That's amazing. So I have work here. No, no.

611
01:00:26,780 --> 01:00:30,059
Okay. What's your name, sir? Of course. Yes. See you.

612
01:00:30,060 --> 01:00:33,350
Won't see you. Can you, auntie? Do you want to say I think. Bruce.

613
01:00:34,310 --> 01:00:39,250
Which Hamilton? Hamilton. Paul Hamilton. Okay.

614
01:00:39,250 --> 01:00:45,990
That's cheating because he's a very famous scientist. You go out to the White House, so so let me do this.

615
01:00:46,000 --> 01:00:51,900
Okay, so k you all t. So the Kurt Hamilton group has just been born.

616
01:00:53,260 --> 01:01:00,010
It's a new symmetrical object built out of the symmetries in the periodic table curtain, so we can come collect this afterwards.

617
01:01:00,490 --> 01:01:05,920
So there are in fact, it's a novel with 25 different surfaces.

618
01:01:05,920 --> 01:01:10,719
Occasionally, it's not one of the atomic symmetries. You can break down two centuries and smaller objects,

619
01:01:10,720 --> 01:01:18,970
but it's amazing that number of metal moves you can do that object which make it sort look like a cube, which is genuinely, distinctly different.

620
01:01:19,300 --> 01:01:27,410
Interestingly, the company that manufactures this Rubik's Cube, the ideal toy company say did all the packaging, the original Rubik's Cubes.

621
01:01:27,430 --> 01:01:32,589
There are more than 3 billion possible states that you can go down now.

622
01:01:32,590 --> 01:01:38,860
Such an underestimate is analogous to McDonald's proudly announcing that they sold more than 120 hamburgers.

623
01:01:39,070 --> 01:01:44,950
So it is obvious that large numbers have now, as you have discovered, many more of these symmetrical objects.

624
01:01:44,950 --> 01:01:51,519
And so you feel that it's a myth that Kurt Hamilton has beaten you to get a bit of immortality.

625
01:01:51,520 --> 01:01:58,179
Actually, we have a project here out of my Muscle Institute where it's for donation to charity that I support.

626
01:01:58,180 --> 01:02:03,340
And in Guatemala, an educational charity which gets kids off the streets and into education.

627
01:02:04,000 --> 01:02:09,159
I will name a smart girl. I'll take one of these new objects that I discovered off.

628
01:02:09,160 --> 01:02:12,730
You are a for a not why you calling but a present for somebody.

629
01:02:13,930 --> 01:02:17,409
Then they're going to own all the money goes to the charity.

630
01:02:17,410 --> 01:02:28,649
But more than just writing a. You know, it was interesting that, you know, we make a lot of progress in my money.

631
01:02:28,650 --> 01:02:30,809
So there's still all of these problems.

632
01:02:30,810 --> 01:02:38,129
And, you know, I think for me, it is those things that, you know, we still don't know, always putting these symmetries together.

633
01:02:38,130 --> 01:02:45,000
We still don't really understand what it is we all get. The monster is in there 196,883 dimensions.

634
01:02:45,000 --> 01:02:52,469
And but for me, that's that's what makes me want to carry on doing mathematics you know broke out killed at the age of 20.

635
01:02:52,470 --> 01:03:03,240
I'm not going to make it off at first, but I often come back to that wonderful quote this Professor Kurokawa told me about a Nico,

636
01:03:04,020 --> 01:03:09,240
which I think sums up mathematics as well as art. You know, in everything, uniformity is undesirable.

637
01:03:09,510 --> 01:03:14,219
There is something interesting and gives. One is the feeling that there is room for growth.

638
01:03:14,220 --> 01:03:18,540
And he says kind of silly on social things which keep us going here.

639
01:03:19,140 --> 01:03:46,879
Study mathematics, I think. So there's an opportunity to ask some questions about symmetry or any other mathematical problems.

640
01:03:46,880 --> 01:03:50,170
You only go on the line what might see.

641
01:03:50,170 --> 01:03:55,640
And so yes. Question in back here we're going to like on the assignments to.

642
01:04:07,670 --> 01:04:11,460
We're saying. All right. We're going to come back to you.

643
01:04:11,790 --> 01:04:17,730
We are going to sit right with that right high.

644
01:04:17,940 --> 01:04:21,750
Is it significant? A maximum number of images there are is a prime number.

645
01:04:24,370 --> 01:04:28,209
Oh, you mean for the Alhambra? Yeah. For example.

646
01:04:28,210 --> 01:04:35,680
Yes. So we were looking at the walls of the Alhambra as as a two dimensional shapes, how they put together.

647
01:04:35,800 --> 01:04:39,940
Interesting that it's 17, which is the final. Yeah, it's a very good question.

648
01:04:39,940 --> 01:04:47,940
If that's the sort of thing that we need to be a mathematician, is that in this case, I think it's not significant actually.

649
01:04:47,950 --> 01:04:51,500
You know, it's my favourite number 17. I mean, I'm obsessed with it.

650
01:04:51,540 --> 01:04:57,310
It's number eight for my football team. And there's the cicadas have a 17 year life cycle.

651
01:04:57,520 --> 01:05:06,820
But actually, I think in this case, actually, 17 is more of a military solution so that you find in the number of ways that you can do things.

652
01:05:06,830 --> 01:05:15,879
It just turns out that 17 is the way because for example, if you go, you can consider sort of Alhambra in higher dimensions as well.

653
01:05:15,880 --> 01:05:24,340
So for example, the number of crystals that are possible is essentially the same question as the crystal structures,

654
01:05:24,340 --> 01:05:27,879
and that isn't so that's the way it has a you know,

655
01:05:27,880 --> 01:05:35,340
it's a good question to ask, but an anxiety, what am I going to prove is it should be climb if I go back and it turns out not to be right.

656
01:05:35,360 --> 01:05:38,500
So it's more trees.

657
01:05:39,550 --> 01:05:45,940
But I see you know that's what arises in mathematicians is asking those looking for those kind of coincidences.

658
01:05:45,940 --> 01:05:50,320
I take the most the most 196,883.

659
01:05:50,620 --> 01:05:53,830
That's the number. And they're saying we got the same number somewhere else.

660
01:05:53,890 --> 01:05:58,510
And I could have just been a coincidence, but when we started to see more numbers appearing,

661
01:05:58,660 --> 01:06:04,690
the next time you see the monster is also hiding inside the ceiling number, there is an explanation.

662
01:06:04,960 --> 01:06:10,330
Then you thought, okay, I'm onto something now. So good question, but that's it.

663
01:06:10,330 --> 01:06:13,500
But after a while we see actually doesn't seem to be significant is.

664
01:06:16,560 --> 01:06:23,760
Who questions is it? We had a question in the middle here and we could get the mike to the guy in the middle, in the blue shirt.

665
01:06:23,760 --> 01:06:27,000
But I'll take this on first. So you just come down with it?

666
01:06:27,240 --> 01:06:32,850
Yes. So you spend a lot of time explaining science and mathematics to people and sharing that knowledge.

667
01:06:33,630 --> 01:06:34,670
Two little questions.

668
01:06:34,860 --> 01:06:41,790
Firstly, there know a push, it seems when we're funded by the public at doing research and that sort of thing, that what we do is useful.

669
01:06:43,290 --> 01:06:48,360
How do you what are your favourite arguments for maintaining mathematical research?

670
01:06:48,930 --> 01:06:51,960
I guess from the very purist level, right through all the scales.

671
01:06:52,170 --> 01:06:59,700
And also, do you think as publicly funded research as should we be doing more of what you're doing,

672
01:06:59,700 --> 01:07:06,000
which is explaining what we do to people and helping them to explain or understand mathematics and its beauty?

673
01:07:06,360 --> 01:07:10,260
Yeah, well, as to the question, I think it's.

674
01:07:12,930 --> 01:07:19,649
We're in a kind of rocky situation with mathematics that very often the research that we do with most mathematicians,

675
01:07:19,650 --> 01:07:26,520
I don't think do mathematics because of utility. We don't do that because we want to create a new piece of technology.

676
01:07:26,700 --> 01:07:31,380
We do mathematics because we feel like we're getting access to internal trees that nature is.

677
01:07:32,130 --> 01:07:36,720
I think aesthetics drives a lot of the directions that we choose in mathematics.

678
01:07:36,720 --> 01:07:44,090
So there's very much more similarity between the arts or very often know the choices that we make in science,

679
01:07:44,120 --> 01:07:48,120
very often driven by trying to explain the physical world around, you know,

680
01:07:48,480 --> 01:07:55,049
these symmetries on discovering things like to see them or why not studying them.

681
01:07:55,050 --> 01:08:03,150
Because I think how we use the stories that I can tell, we are all magical and full of surprises and excite me.

682
01:08:03,150 --> 01:08:08,310
And that's and a sense of your second question, which is why why do I do this?

683
01:08:08,760 --> 01:08:16,800
I do this partly because I want the next generation of mathematicians to and I want to excite them.

684
01:08:16,920 --> 01:08:25,049
My I'm paying back my teachers to Alison. He took me out of the class in the middle of a lesson and got me excited about mathematics.

685
01:08:25,050 --> 01:08:28,500
And I sort of want to pay my little way for saying, okay,

686
01:08:28,760 --> 01:08:35,800
I want to tell people as well so they can get excited and become mathematicians as well as she's selfish.

687
01:08:36,510 --> 01:08:42,340
The reason I do this because I love my subject so much, I love talking about it and then communicate, communicating.

688
01:08:42,840 --> 01:08:47,440
And for me, I really when I make a mathematical discovery, there are two things about being in a position.

689
01:08:47,490 --> 01:08:52,990
One is making a new discovery, but the second is communicating it, and that's when it starts to come online.

690
01:08:53,010 --> 01:08:58,020
When I tell you about something I've done or I'm in a seminar here in the department,

691
01:08:58,050 --> 01:09:01,290
or I write a paper and I send it to a journal and somebody else reads it.

692
01:09:01,530 --> 01:09:04,950
That's the moment when suddenly the thing has life and it begins to breathe.

693
01:09:04,950 --> 01:09:11,630
And and so I think that's, you know, we all do this in some sort of level.

694
01:09:11,640 --> 01:09:16,830
And the more we communicate, the more that subject lives and breathes.

695
01:09:17,670 --> 01:09:19,409
But I think we all have these opportunities,

696
01:09:19,410 --> 01:09:25,650
although a lot of these symmetrical objects we discovered purely for the excitement of discovering something new,

697
01:09:26,010 --> 01:09:31,229
actually, we can very quickly connect it with a utility.

698
01:09:31,230 --> 01:09:34,800
And the fact that this can create new technology,

699
01:09:35,640 --> 01:09:43,290
for example in the is the Acme supply and I simple said was an object is just discovered in use imagery which actually

700
01:09:43,290 --> 01:09:52,259
that became the basis of a new Eric Watson code that was used by Voyager to take pictures of Saturn and these pictures,

701
01:09:52,260 --> 01:09:54,420
right. Corrupted when they were delivered back to Earth.

702
01:09:54,630 --> 01:10:01,230
But using the symmetry of this code, they were able to correct these errors and produce some crystal clear images.

703
01:10:01,860 --> 01:10:07,950
That is a lot like if I give an article and experience in the future, but this is my error in one corner.

704
01:10:08,160 --> 01:10:11,890
Rushing Roshi know how to correct error because of the symmetry and the rest of the target.

705
01:10:12,270 --> 01:10:20,249
So symmetry actually is at the heart of many of the photos which encode a picture or a voice in the digital data,

706
01:10:20,250 --> 01:10:22,590
which tend to get to be corrupted in on the way back.

707
01:10:23,370 --> 01:10:33,350
So I think we can very easily justify why nobody discovered that symmetry because they wanted to create the perfect prime numbers.

708
01:10:33,360 --> 01:10:40,679
I used the Internet cryptography, but the things we discovered by our numbers that I use were discovered by Fama and Weiner.

709
01:10:40,680 --> 01:10:46,649
They to know about the Internet. But I think it's a balance.

710
01:10:46,650 --> 01:10:52,710
And, you know, I recognise that governments have limited resources, but you must always have a balance for value.

711
01:10:53,520 --> 01:10:59,130
And so yes, sure, there are some things which we can see have short term goals and there's always other.

712
01:10:59,150 --> 01:11:03,510
But there are many things that's that's look at this.

713
01:11:03,510 --> 01:11:07,920
I will have another thing that might be useful very much later.

714
01:11:08,010 --> 01:11:13,440
Turned out to be just the thing you need to do to to Google, for example.

715
01:11:13,440 --> 01:11:19,110
Google isn't like Google elves just sitting there looking at the websites that you've just asked about.

716
01:11:19,320 --> 01:11:26,160
It's a clever bit of mathematics, eigenvalues of matrices, which helps you to set up exactly what you're looking for.

717
01:11:26,820 --> 01:11:38,430
So it's, I think, through the way that we should justify why still resources research our brains about symmetry should be funded,

718
01:11:38,430 --> 01:11:40,290
even though we may not know where it's going to end.

719
01:11:40,290 --> 01:11:48,200
Is that huge number of examples of mathematical research that has been done which which has been subsequently catalysed.

720
01:11:48,480 --> 01:11:56,760
But I think we shouldn't miss the fact that actually still worth as a as a culture, you know, why should we come to music?

721
01:11:57,120 --> 01:12:04,590
You know, we we are for our society. If we don't have these different strands, each one helps the other.

722
01:12:04,680 --> 01:12:09,030
I think, you know, I spend a lot of time talking with artists.

723
01:12:10,230 --> 01:12:14,310
I'm doing a project of the moments with the theatre company. The House mathematics.

724
01:12:14,600 --> 01:12:19,650
Actually, that helps me because they make you look at a question in a completely different way and have a different way of looking at things.

725
01:12:19,830 --> 01:12:23,730
And that's, you know, I really enjoy it because creating a piece of paper about maths.

726
01:12:23,880 --> 01:12:27,930
But actually it's, I think it benefits all of these things enriching each other.

727
01:12:27,930 --> 01:12:31,270
And if you count one else, you are going to have a much poorer society.

728
01:12:33,600 --> 01:12:35,730
And so the question here.

729
01:12:38,490 --> 01:12:47,640
I probably some of the other people noticed as they came into this building outside one of the doors there's that creating a tiled surface.

730
01:12:47,790 --> 01:12:50,790
Penrose tiles based on an Oxford discovery.

731
01:12:51,210 --> 01:12:57,780
And they're marked with lines, curving lines. And some of those lines make a neat circle, about six feet in diameter.

732
01:12:58,290 --> 01:13:03,090
But while I was waiting to come in, I tried following the lines like a labyrinth.

733
01:13:03,780 --> 01:13:07,470
Some of the more complex patterns, like the outline of a five bedroom floor.

734
01:13:08,160 --> 01:13:10,350
But one line went on and on and on.

735
01:13:10,560 --> 01:13:18,810
And I was wondering, is that potentially an infinite line in that part of the layout of the tiles isn't complete yet,

736
01:13:18,810 --> 01:13:22,080
but I'm afraid it's going to be fine. It's not like that.

737
01:13:22,300 --> 01:13:31,290
Well, this is the beautiful thing about that design, is that it's increasing its symmetry,

738
01:13:31,290 --> 01:13:38,490
like five pointed star or something, but it has no repeating symmetry at all.

739
01:13:38,520 --> 01:13:46,760
So now I'm I would have to do a little bit of research to see whether there is an infinite line inside there.

740
01:13:46,770 --> 01:13:50,250
But I'm absolutely sure that there will be an unbounded.

741
01:13:50,310 --> 01:13:58,140
So the even if they're will find it will definitely be an unbounded number, I suspect, and I won't.

742
01:13:58,500 --> 01:14:04,050
But the interesting, interesting question and, you know, if given this answer and with all their actual policies parsing,

743
01:14:04,500 --> 01:14:11,250
which must be infinite and what what you discovered with this tiny is a very interesting time because in some sense it

744
01:14:12,060 --> 01:14:19,260
contradicts what I said about the Alhambra because it looks like an 18 mean you taught it but it doesn't have the symmetry.

745
01:14:19,260 --> 01:14:25,169
It has no move you can make inside those tiles such that if you move in some way,

746
01:14:25,170 --> 01:14:31,290
they would sit back on top of each other because of what is known periodicity to it so dramatically.

747
01:14:31,290 --> 01:14:35,010
They're objects that have symmetry that by force, symmetry or other symmetries,

748
01:14:35,220 --> 01:14:42,840
you were able to prove the global act on water treaties that nature actually discovered these before Roger.

749
01:14:43,560 --> 01:14:54,660
Roger actually famously and his family discovered this pattern of been used on some toilet paper.

750
01:14:54,670 --> 01:14:59,190
And this is very useful for toilet paper because because of the Non-repeating passage,

751
01:14:59,190 --> 01:15:02,820
you don't get the kind of build up if you wrap your mind around, around.

752
01:15:02,850 --> 01:15:06,060
So you have some experience, you can get a nasty sort of build.

753
01:15:06,210 --> 01:15:11,500
So this company had used it had actually cleverly patented the design.

754
01:15:11,520 --> 01:15:18,090
And I think they settled out of court for some time.

755
01:15:18,690 --> 01:15:23,940
But actually it's be discovered before Roger because we found crystals, quality crystals.

756
01:15:24,390 --> 01:15:29,970
One of the recent Nobel Prizes in chemistry was for the investigation of these crystals,

757
01:15:30,300 --> 01:15:36,180
which is quite remarkable because it is when you start to make the tiles.

758
01:15:36,180 --> 01:15:36,389
I mean,

759
01:15:36,390 --> 01:15:45,709
they have been very careful outside because there's you sort of have to have global knowledge for the whole structure to be able to lay it down.

760
01:15:45,710 --> 01:15:48,090
And so when a crystal is forming these quasicrystals,

761
01:15:48,750 --> 01:15:57,750
it's quite an interesting question about how does out how does this kind of crystal known global properties, although it's to down as it grows.

762
01:15:58,760 --> 01:16:07,480
So there are a lot of interesting questions around the this norm here only can I say around including on the US.

763
01:16:18,320 --> 01:16:26,750
Do you believe mathematics is discovered and created? And because I know Girdle said that axiom to something we believe which is true,

764
01:16:26,750 --> 01:16:32,150
but nowadays people call things which are axioms, which we don't necessarily know whether they're true or not.

765
01:16:32,450 --> 01:16:36,190
I do believe the truth lies in mathematics all the time.

766
01:16:36,200 --> 01:16:43,670
Or maybe the hypothesis is that we haven't breathed, which I actually don't have any proof, like the continuum hypothesis.

767
01:16:43,830 --> 01:16:45,350
Yeah, well,

768
01:16:45,350 --> 01:16:54,710
I think that it's it's a kind of eternal tension which exists in a mathematician's life about these two words about discovery and creation.

769
01:16:56,510 --> 01:17:00,380
And I find myself using both of them at different times.

770
01:17:00,650 --> 01:17:05,629
For example, there is new symmetrical objects now.

771
01:17:05,630 --> 01:17:12,900
I felt I created that. I felt that it was an act of creation to somehow piece together these new set of

772
01:17:12,920 --> 01:17:18,350
ideas to create this shape which has lovely connections and with electric cars.

773
01:17:18,890 --> 01:17:27,200
And there was a time I had this feeling of clashing that was all that was there all along, somewhat easier to put together and discover.

774
01:17:28,100 --> 01:17:30,280
So then you start to it like discovery.

775
01:17:30,290 --> 01:17:39,619
And I don't think there is this continual tension about how much and you see this in a lot of mathematical discoveries gets discovered.

776
01:17:39,620 --> 01:17:44,480
At the same time, non-Euclidean geometry was discovered by three different people.

777
01:17:44,720 --> 01:17:50,360
So it's a field like it was it was a continent out there for mathematicians to discover.

778
01:17:50,360 --> 01:17:56,630
And it was Gauss Mollaei who who both came close together.

779
01:17:58,040 --> 01:18:02,959
And, you know, you want to say that about and I did The Magic Flute by Mozart.

780
01:18:02,960 --> 01:18:06,080
You wouldn't think that anybody else could create the Magic Flute.

781
01:18:06,370 --> 01:18:16,699
It really isn't that isn't a discovery. But actually, if you talk to a lot of composers, you find them actually countering that and saying,

782
01:18:16,700 --> 01:18:24,439
well, you'd be surprised how often actually a new musical idea actually has been zeitgeisty.

783
01:18:24,440 --> 01:18:28,820
And people come across the same musical idea almost simultaneously.

784
01:18:29,960 --> 01:18:34,970
They would write the same piece of music. But then even in mathematics, we ratti we might have similar ideas.

785
01:18:34,970 --> 01:18:37,790
We rarely write exactly the same mathematics.

786
01:18:38,340 --> 01:18:45,560
And so I think actually even in the arts, one can see a little bit more of a tension between this creativity and discovery.

787
01:18:45,920 --> 01:18:46,700
There's interesting story.

788
01:18:46,700 --> 01:18:56,510
I was at a festival with Michael Nyman and I asked him a question during the session about whether what did he think as a composer,

789
01:18:56,690 --> 01:19:01,160
whether he thought music was just created, or was there an element of.

790
01:19:01,490 --> 01:19:05,990
Do you feel like you ever discovered music? And he said, Well, yes, I can give you an example of this.

791
01:19:06,740 --> 01:19:14,360
And he said, I found this amazing sequence, of course, where you can get a soprano to just hold a note all the way through.

792
01:19:14,360 --> 01:19:20,470
The chords just changed around her, but she just kept this one note and it was a very beautiful sort of property.

793
01:19:20,480 --> 01:19:23,330
So the sequence, of course, and he was very proud of it.

794
01:19:23,780 --> 01:19:29,750
And then a few years later, he bought an album by the Scissor Sisters, and it was an extra material.

795
01:19:29,890 --> 01:19:32,000
It was an encore from an article or something.

796
01:19:32,330 --> 01:19:38,600
And you listen to some of his material and he heard the same sequence, of course, what, running through one of the songs.

797
01:19:39,230 --> 01:19:44,330
And he said he couldn't believe that they'd found it independently.

798
01:19:44,360 --> 01:19:48,500
So yeah, she took them to court for ripping off his music.

799
01:19:49,190 --> 01:19:51,890
And so there was a whole debate within the court case,

800
01:19:52,040 --> 01:19:56,900
and they got musicologists to try and see whether the sequence of chords and ever be found before.

801
01:19:57,230 --> 01:20:03,830
And eventually Scissor Sisters had to settle and email that they actually had written off because there was no evidence that.

802
01:20:04,250 --> 01:20:13,760
Michael Nyman take that as an example. Actually, music, you can make discoveries of structures in a way that you think is just creative.

803
01:20:14,240 --> 01:20:20,840
Now coming to a question about Google is very interesting because that fact gave us a little bit of room for creativity,

804
01:20:21,110 --> 01:20:30,950
because you mentioned the continuum hypothesis. You know, we which I will actually come to in one of my other lectures later on.

805
01:20:31,700 --> 01:20:39,560
But actually it shows that we got this question about whether there's a set of numbers which is

806
01:20:39,710 --> 01:20:46,100
strictly bigger than the whole numbers and strictly smaller than the rational numbers of real numbers.

807
01:20:46,550 --> 01:20:50,770
And it turns out that we can make a choice. It's a bit like the answer is yes and no.

808
01:20:50,960 --> 01:20:53,510
I think it's what students would love in an exam. All right.

809
01:20:55,460 --> 01:21:02,840
So I think it's a very deep philosophical question you are answering, which I think we we constantly wrangle with in a way,

810
01:21:02,840 --> 01:21:06,560
how many stick our head in the round about because we just go on doing what we do.

811
01:21:10,510 --> 01:21:14,960
That's. You sound a.

812
01:21:31,190 --> 01:21:34,380
I'll try and talk loudly as it's coming through. Brilliant.

813
01:21:34,400 --> 01:21:38,710
Excellent. I love to see the pictures that have burned already.

814
01:21:38,720 --> 01:21:48,290
Scale of the Alhambra. And it seems absolutely incredible that they got all 17 symmetry elements in two days in the Alhambra.

815
01:21:49,100 --> 01:21:52,999
I've been that. I'm afraid I read as much about it as I would have liked to.

816
01:21:53,000 --> 01:21:58,200
But do you know? Well, so did the people who designed the Alhambra.

817
01:21:58,250 --> 01:22:03,620
Have they done a lot of I guess that they looked around in nature and kind of looked at the structures and nature?

818
01:22:03,620 --> 01:22:04,729
I mean, does it it can't.

819
01:22:04,730 --> 01:22:11,450
It doesn't it feels like it can't be a coincidence or is it something about the sort of symmetry elements are almost wired into us in some way,

820
01:22:11,450 --> 01:22:18,200
I guess is a slightly psychological question to put to you rather mathematical one, but I'm really interested to hear your thoughts on that.

821
01:22:19,660 --> 01:22:27,160
I think that's a that I think we all very hard line to be extremely sensitive to symmetry.

822
01:22:27,200 --> 01:22:32,440
I mean I think at that which is why we find it again and again,

823
01:22:32,560 --> 01:22:41,410
it's something we play with or withdraw and there's lots of evidence of the brain is be so addicted to symmetry,

824
01:22:41,410 --> 01:22:49,090
the desire to see symmetry where there isn't a relation to those inkblots, you know,

825
01:22:49,090 --> 01:22:57,100
there's able to there's a basically is just a passive which is the god symmetry symmetry around it and use those because.

826
01:22:58,920 --> 01:23:04,620
He knew that we would try and interpret. And if you're in a psychological session, you will try and tell a story.

827
01:23:04,860 --> 01:23:09,920
It is called symmetry. It's me telling you something. And so you started to unleash your inner mind.

828
01:23:09,960 --> 01:23:13,110
You know that it causes no. Got anything particularly special about it.

829
01:23:13,110 --> 01:23:23,060
But these different rules of images just seem to tease out of you as stories because you try and interpret it well.

830
01:23:23,100 --> 01:23:26,660
You used to get all his patients to draw mandibles,

831
01:23:26,880 --> 01:23:33,510
these symmetrical kind of Buddhist shapes because he believed in they would express their inner workings of their mind.

832
01:23:33,510 --> 01:23:36,180
So as soon as Freud said, I see blank,

833
01:23:36,180 --> 01:23:45,089
I believe in a lot of psychological problems based on the fact that we try to make logic symmetrical logic is very asymmetrical.

834
01:23:45,090 --> 01:23:51,600
If the if A and B, that's what he does not be repeating that you want to reverse these things.

835
01:23:52,470 --> 01:24:00,030
But actually the mindset is always to reverse the thing that actually that could be responsible for strange psychological disorders,

836
01:24:01,020 --> 01:24:10,670
for example, is was a patient of his. That's when when blood was being taken from the arm was that the arm was being removed from the bodies.

837
01:24:11,040 --> 01:24:19,620
And just to this reversal, and if you look at a lot of things that you can find the same sort of explanations that this sort of rape is going on.

838
01:24:19,630 --> 01:24:23,540
So I think going to the psychological side, I think that, you know,

839
01:24:23,590 --> 01:24:29,750
the brain is so online that sometimes it can confuse itself and, you know, hungers.

840
01:24:29,760 --> 01:24:37,110
I think because I think this is a good example. I think all of those images were things that they would have naturally seen around them.

841
01:24:37,200 --> 01:24:46,350
I think this is a great example of, first of all, being inspired by nature and a lot of the questions of science and the like is stalled in nature.

842
01:24:46,560 --> 01:24:55,080
But then we lose our imaginations, run riots, and and we start to try and push things which are beyond the natural world.

843
01:24:55,080 --> 01:24:58,950
And a lot of the mathematics, I think you can find its source in the natural world,

844
01:24:58,950 --> 01:25:03,420
but then it's about trying to find something and the other possibilities.

845
01:25:03,780 --> 01:25:11,219
So I would say that some of the Moorish artists who came up within the 14th century, those designs,

846
01:25:11,220 --> 01:25:18,840
and it's interesting that I showed you those patterns with the same symmetries, although they look very different.

847
01:25:19,440 --> 01:25:22,920
Interestingly, you find those in the same bit of the palace.

848
01:25:23,820 --> 01:25:29,070
So the entrance to the palace is full of lots of rotations of the floor.

849
01:25:29,110 --> 01:25:35,429
Two reflections and the second floor of the House is where I was and people visiting

850
01:25:35,430 --> 01:25:40,530
the palace would be hosting it and it's quite clients and you could see the symmetry,

851
01:25:40,530 --> 01:25:47,069
right. Some of the symmetries. Well the different patterns as you go through into the more intimate parts of the palace to the harem,

852
01:25:47,070 --> 01:25:53,910
for example, then you suddenly see reflections disappearing. You have triangles, hexagons, you get rotations.

853
01:25:54,300 --> 01:26:01,590
So a different sort of symmetry. But it seems like they always had a sensitivity to the fact that they they felt like these had some connection,

854
01:26:01,740 --> 01:26:06,990
although they never had the language until the 19th century. For us to be able to say, why are these?

855
01:26:06,990 --> 01:26:11,310
Why do you feel like these should be in the same bit of the past? But I think it is.

856
01:26:11,790 --> 01:26:14,990
That's why it comes back to the other question.

857
01:26:15,000 --> 01:26:19,139
So I think having this combination of ways of looking at the world,

858
01:26:19,140 --> 01:26:26,580
why having an artistic sentiments as a mathematician is a very helpful one in creating new possibilities.

859
01:26:27,030 --> 01:26:32,260
And in a way you see the artistic side running ahead of the mathematical side in the Alhambra.

860
01:26:37,490 --> 01:26:52,129
Is there another question here? I just think, yeah, I just study fractals for school.

861
01:26:52,130 --> 01:27:02,300
And I was wondering, does it make any sense and is it worth studying symmetry in a real value dimension of that's great to do because you want say,

862
01:27:02,300 --> 01:27:08,060
well, fractals are kind of the most magical thing because of all that's kind of strange.

863
01:27:08,180 --> 01:27:13,010
But actually there's a symmetry, a different sort of symmetry happening here, which is a symmetry scale.

864
01:27:13,880 --> 01:27:16,700
So that's another move that you could make use of.

865
01:27:16,700 --> 01:27:23,359
Zoom in and then suddenly you see the same thing repeating itself and then you get this infinite repetition of the fractal.

866
01:27:23,360 --> 01:27:32,580
So that's actually symmetry is a very powerful tool in seeing in areas which doesn't look like it should have any sort of impact for,

867
01:27:32,600 --> 01:27:38,120
say, symmetry in the world. Or perhaps the geometry is still very powerful.

868
01:27:38,510 --> 01:27:49,240
So, yes. How do people go about attacking problems like.

869
01:27:49,600 --> 01:27:55,600
Trying to classify all of the potential building blocks for symmetries and discovering things like this monster group,

870
01:27:55,600 --> 01:27:59,380
which is certainly not an object that you just trip on out in the street or walking around.

871
01:27:59,590 --> 01:28:04,540
Yeah, that's a that is one of the eternal mysteries.

872
01:28:04,540 --> 01:28:09,040
And when you're a research mathematician, I wish I knew the answer what the solution is.

873
01:28:09,220 --> 01:28:15,610
And it is about playing around with these and any kind of and it's kind of intriguing that there

874
01:28:15,610 --> 01:28:20,110
were periods in trying to understand the classification of the building blocks of symmetry.

875
01:28:20,380 --> 01:28:27,220
What I think people just had the impression this is never going to end, we're never going to find all of the things.

876
01:28:27,220 --> 01:28:35,470
And we you know, perhaps this is just beyond the capacity of the human brain, even when flying across the globe of the whole community.

877
01:28:37,040 --> 01:28:40,520
But as you put more and more restrictions on I mean, the monster.

878
01:28:40,720 --> 01:28:44,200
Exactly. It's it's a little bit like fundamental particles.

879
01:28:44,200 --> 01:28:50,360
The monsters was predicted because of conditions that it had before it was ever constructed.

880
01:28:50,380 --> 01:28:57,040
So there was a thing like that should be an object with these particular properties because we can't find any reason why they shouldn't be.

881
01:28:57,250 --> 01:29:04,930
And so the conjecture, they were such that somebody conjectured there should be an object with this number of symmetries and certain properties.

882
01:29:05,210 --> 01:29:08,020
And the challenge was, okay, well, let's try and build this thing.

883
01:29:08,400 --> 01:29:18,280
And if we can use a very complex object, yet at its heart is what do you need when you're doing these properties to break down into smaller pieces?

884
01:29:18,520 --> 01:29:22,540
And the smaller piece in the case of the monster was what you might say is not smaller.

885
01:29:22,560 --> 01:29:29,980
That and there's a way to pack 24 dimensional oranges, something called the beach lattice, which we knew about.

886
01:29:29,980 --> 01:29:33,400
And the centrepiece of that we started is it gave us the way to build.

887
01:29:33,670 --> 01:29:40,870
So you sort of these smaller jigsaw bits very often to help you to piece together to make a larger story.

888
01:29:41,470 --> 01:29:53,440
But in the case of the story of symmetry, it's a wonderful example out of many hundreds of mathematicians were involved in sort of exploring this,

889
01:29:53,450 --> 01:29:59,529
it was a bit like going out and exploring, finding new continents for archipelago of islands,

890
01:29:59,530 --> 01:30:09,670
which seem to have nothing to do with anything that your home actually has a vague thought that maybe we we didn't think we found.

891
01:30:10,150 --> 01:30:14,440
But he says we could easily have some or will have on these estimates.

892
01:30:14,440 --> 01:30:17,230
Also, he thinks that there might be a chance,

893
01:30:17,890 --> 01:30:27,810
there might be something out there that we that we miss because we thought our conditions that it couldn't be satisfied, which actually they could.

894
01:30:27,820 --> 01:30:31,870
So he has a theory, which is that I mean, you know, it's a strange thing,

895
01:30:32,530 --> 01:30:37,180
this sort of tension between wanting to prove the conjecture you're working on.

896
01:30:37,390 --> 01:30:45,400
But when he proved it, it's a sort of melancholy because, you know, you've been with that in conjecture for so long.

897
01:30:45,400 --> 01:30:48,860
And I think when Fermat's Last Theorem proved and we all love.

898
01:30:48,860 --> 01:30:56,600
And was he finished? No, but there was another guy. And by this I only realise that that's not such a motivating problem for us.

899
01:30:57,370 --> 01:31:02,950
It was kind of a know, I think it was a moment of sadness in life, wonderful cracks.

900
01:31:03,280 --> 01:31:10,480
And there was the same symmetry. It was like, Oh gosh, let's do more questions that.

901
01:31:15,710 --> 01:31:22,070
I think we should finish there. It's not what you wants to do tomorrow, so thank you for doing great.

902
01:31:37,410 --> 01:31:55,860
Sorry. You only had one.

903
01:31:56,260 --> 01:31:59,770
Yeah, it was just. Hello? I'm not sure where to start.

904
01:31:59,800 --> 01:32:04,450
Where we got me, actually. Uh, yeah.

905
01:32:04,690 --> 01:32:07,720
Just seeing Charlie. Yeah. Yeah, I just. I just took our.

906
01:32:37,250 --> 01:32:44,410
Oh, that is sound. No.

907
01:32:44,670 --> 01:33:00,050
Because you. It's.

908
01:33:03,860 --> 01:33:17,140
And I tried. On the 30th of May 1832, a gunshot was heard ringing out across the 30 on this one in Paris.

909
01:33:17,650 --> 01:33:19,780
Apparently it was walking to market that morning,

910
01:33:19,780 --> 01:33:28,590
ran towards where the gunshot came from and on the ground and found a young man arriving in a garden, clearly killed by a shooting range.

911
01:33:29,020 --> 01:33:34,719
The young man's name was every single gunman was taken to the local hospital,

912
01:33:34,720 --> 01:33:39,490
the coaching hospital where he died the next morning in the arms of his brother Alfred.

913
01:33:39,580 --> 01:33:43,840
And he said to his brother on his death, he says, Don't cry from me, Alfred.

914
01:33:43,840 --> 01:33:48,190
I need all the courage I can muster to die at the age of 20.

915
01:33:49,150 --> 01:33:52,270
I hope likely will at you was about that morning.

916
01:33:52,960 --> 01:33:57,750
It must have been between Valois and a friend of his over a woman that they were both in love with them.

917
01:33:57,890 --> 01:34:04,150
Others suggested that, in fact, it was the establishment trying to get rid of this troublesome revolutionary.

918
01:34:04,780 --> 01:34:11,230
There's even a conjecture that Galois staged his own dance in order to try and spark a new revolution in Paris.

919
01:34:11,800 --> 01:34:14,980
What happened to Clay? What statue was about that morning?

920
01:34:15,170 --> 01:34:22,450
This actually wasn't revolutionary politics, which Galois was famous because a few years earlier, while still at school.

921
01:34:22,840 --> 01:34:31,669
That was so one of the great unsolved problems in mathematics versions of the academicians in Paris trying to explain his great breakthrough.

922
01:34:31,670 --> 01:34:38,950
But unfortunately, the academicians couldn't understand what on earth he was writing about, only because this is how he wrote most of his mathematics.

923
01:34:39,760 --> 01:34:41,680
But anyway, the night before that June,

924
01:34:41,980 --> 01:34:48,080
I realised that this probably was the last opportunity he was going to have to try and explain his great breakthrough.

925
01:34:48,100 --> 01:34:52,659
So he stayed up all night writing in a letter to a friend trying to articulate this great

926
01:34:52,660 --> 01:34:58,990
breakthrough as the Sun appeared over the new day and he went out to meet his destiny.

927
01:34:59,320 --> 01:35:03,620
Maybe the fact that he was doing maths all night, the fact that he was such a bad shot that morning.

928
01:35:03,660 --> 01:35:08,229
But actually, those documents that he left behind on the desk,

929
01:35:08,230 --> 01:35:13,570
which many of us regard as some of the most important documents in the history of mathematics,

930
01:35:13,570 --> 01:35:16,750
because hidden inside, there was a new language,

931
01:35:16,990 --> 01:35:25,690
a language to understand one of the most important concepts in mathematics and science, namely the subject of symmetry.

932
01:35:25,960 --> 01:35:28,390
This new language was called group theory.