Marcus du Sautoy: Symmetry, reality's riddle

131,456 views ・ 2009-10-29

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00:18
On the 30th of May, 1832,
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a gunshot was heard
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ringing out across the 13th arrondissement in Paris.
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(Gunshot)
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A peasant, who was walking to market that morning,
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ran towards where the gunshot had come from,
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and found a young man writhing in agony on the floor,
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clearly shot by a dueling wound.
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The young man's name was Evariste Galois.
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He was a well-known revolutionary in Paris at the time.
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Galois was taken to the local hospital
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where he died the next day in the arms of his brother.
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And the last words he said to his brother were,
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"Don't cry for me, Alfred.
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I need all the courage I can muster
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to die at the age of 20."
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It wasn't, in fact, revolutionary politics
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for which Galois was famous.
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But a few years earlier, while still at school,
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he'd actually cracked one of the big mathematical
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problems at the time.
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And he wrote to the academicians in Paris,
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trying to explain his theory.
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But the academicians couldn't understand anything that he wrote.
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(Laughter)
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This is how he wrote most of his mathematics.
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So, the night before that duel, he realized
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this possibly is his last chance
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to try and explain his great breakthrough.
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So he stayed up the whole night, writing away,
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trying to explain his ideas.
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And as the dawn came up and he went to meet his destiny,
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he left this pile of papers on the table for the next generation.
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Maybe the fact that he stayed up all night doing mathematics
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was the fact that he was such a bad shot that morning and got killed.
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But contained inside those documents
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was a new language, a language to understand
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one of the most fundamental concepts
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of science -- namely symmetry.
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Now, symmetry is almost nature's language.
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It helps us to understand so many
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different bits of the scientific world.
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For example, molecular structure.
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What crystals are possible,
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we can understand through the mathematics of symmetry.
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In microbiology you really don't want to get a symmetrical object,
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because they are generally rather nasty.
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The swine flu virus, at the moment, is a symmetrical object.
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And it uses the efficiency of symmetry
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to be able to propagate itself so well.
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But on a larger scale of biology, actually symmetry is very important,
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because it actually communicates genetic information.
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I've taken two pictures here and I've made them artificially symmetrical.
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And if I ask you which of these you find more beautiful,
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you're probably drawn to the lower two.
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Because it is hard to make symmetry.
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And if you can make yourself symmetrical, you're sending out a sign
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that you've got good genes, you've got a good upbringing
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and therefore you'll make a good mate.
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So symmetry is a language which can help to communicate
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genetic information.
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Symmetry can also help us to explain
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what's happening in the Large Hadron Collider in CERN.
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Or what's not happening in the Large Hadron Collider in CERN.
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To be able to make predictions about the fundamental particles
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we might see there,
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it seems that they are all facets of some strange symmetrical shape
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in a higher dimensional space.
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And I think Galileo summed up, very nicely,
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the power of mathematics
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to understand the scientific world around us.
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He wrote, "The universe cannot be read
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until we have learnt the language
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and become familiar with the characters in which it is written.
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It is written in mathematical language,
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and the letters are triangles, circles and other geometric figures,
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without which means it is humanly impossible
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to comprehend a single word."
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But it's not just scientists who are interested in symmetry.
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Artists too love to play around with symmetry.
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They also have a slightly more ambiguous relationship with it.
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Here is Thomas Mann talking about symmetry in "The Magic Mountain."
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He has a character describing the snowflake,
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and he says he "shuddered at its perfect precision,
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found it deathly, the very marrow of death."
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But what artists like to do is to set up expectations
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of symmetry and then break them.
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And a beautiful example of this
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I found, actually, when I visited a colleague of mine
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in Japan, Professor Kurokawa.
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And he took me up to the temples in Nikko.
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And just after this photo was taken we walked up the stairs.
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And the gateway you see behind
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has eight columns, with beautiful symmetrical designs on them.
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Seven of them are exactly the same,
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and the eighth one is turned upside down.
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And I said to Professor Kurokawa,
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"Wow, the architects must have really been kicking themselves
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when they realized that they'd made a mistake and put this one upside down."
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And he said, "No, no, no. It was a very deliberate act."
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And he referred me to this lovely quote from the Japanese
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"Essays in Idleness" from the 14th century,
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in which the essayist wrote, "In everything,
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uniformity is undesirable.
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Leaving something incomplete makes it interesting,
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and gives one the feeling that there is room for growth."
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Even when building the Imperial Palace,
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they always leave one place unfinished.
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But if I had to choose one building in the world
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to be cast out on a desert island, to live the rest of my life,
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being an addict of symmetry, I would probably choose the Alhambra in Granada.
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This is a palace celebrating symmetry.
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Recently I took my family --
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we do these rather kind of nerdy mathematical trips, which my family love.
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This is my son Tamer. You can see
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he's really enjoying our mathematical trip to the Alhambra.
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But I wanted to try and enrich him.
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I think one of the problems about school mathematics
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is it doesn't look at how mathematics is embedded
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in the world we live in.
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So, I wanted to open his eyes up to
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how much symmetry is running through the Alhambra.
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You see it already. Immediately you go in,
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the reflective symmetry in the water.
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But it's on the walls where all the exciting things are happening.
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The Moorish artists were denied the possibility
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to draw things with souls.
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So they explored a more geometric art.
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And so what is symmetry?
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The Alhambra somehow asks all of these questions.
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What is symmetry? When [there] are two of these walls,
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do they have the same symmetries?
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Can we say whether they discovered
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all of the symmetries in the Alhambra?
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And it was Galois who produced a language
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to be able to answer some of these questions.
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For Galois, symmetry -- unlike for Thomas Mann,
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which was something still and deathly --
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for Galois, symmetry was all about motion.
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What can you do to a symmetrical object,
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move it in some way, so it looks the same
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as before you moved it?
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I like to describe it as the magic trick moves.
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What can you do to something? You close your eyes.
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I do something, put it back down again.
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It looks like it did before it started.
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So, for example, the walls in the Alhambra --
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I can take all of these tiles, and fix them at the yellow place,
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rotate them by 90 degrees,
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put them all back down again and they fit perfectly down there.
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And if you open your eyes again, you wouldn't know that they'd moved.
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But it's the motion that really characterizes the symmetry
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inside the Alhambra.
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But it's also about producing a language to describe this.
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And the power of mathematics is often
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to change one thing into another, to change geometry into language.
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So I'm going to take you through, perhaps push you a little bit mathematically --
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so brace yourselves --
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push you a little bit to understand how this language works,
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which enables us to capture what is symmetry.
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So, let's take these two symmetrical objects here.
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Let's take the twisted six-pointed starfish.
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What can I do to the starfish which makes it look the same?
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Well, there I rotated it by a sixth of a turn,
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and still it looks like it did before I started.
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I could rotate it by a third of a turn,
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or a half a turn,
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or put it back down on its image, or two thirds of a turn.
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And a fifth symmetry, I can rotate it by five sixths of a turn.
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And those are things that I can do to the symmetrical object
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that make it look like it did before I started.
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Now, for Galois, there was actually a sixth symmetry.
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Can anybody think what else I could do to this
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which would leave it like I did before I started?
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I can't flip it because I've put a little twist on it, haven't I?
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It's got no reflective symmetry.
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But what I could do is just leave it where it is,
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pick it up, and put it down again.
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And for Galois this was like the zeroth symmetry.
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Actually, the invention of the number zero
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was a very modern concept, seventh century A.D., by the Indians.
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It seems mad to talk about nothing.
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And this is the same idea. This is a symmetrical --
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so everything has symmetry, where you just leave it where it is.
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So, this object has six symmetries.
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And what about the triangle?
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Well, I can rotate by a third of a turn clockwise
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or a third of a turn anticlockwise.
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But now this has some reflectional symmetry.
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I can reflect it in the line through X,
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or the line through Y,
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or the line through Z.
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Five symmetries and then of course the zeroth symmetry
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where I just pick it up and leave it where it is.
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So both of these objects have six symmetries.
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Now, I'm a great believer that mathematics is not a spectator sport,
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and you have to do some mathematics
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in order to really understand it.
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So here is a little question for you.
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And I'm going to give a prize at the end of my talk
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for the person who gets closest to the answer.
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The Rubik's Cube.
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How many symmetries does a Rubik's Cube have?
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How many things can I do to this object
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and put it down so it still looks like a cube?
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Okay? So I want you to think about that problem as we go on,
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and count how many symmetries there are.
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And there will be a prize for the person who gets closest at the end.
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But let's go back down to symmetries that I got for these two objects.
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What Galois realized: it isn't just the individual symmetries,
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but how they interact with each other
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which really characterizes the symmetry of an object.
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If I do one magic trick move followed by another,
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the combination is a third magic trick move.
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And here we see Galois starting to develop
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a language to see the substance
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of the things unseen, the sort of abstract idea
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of the symmetry underlying this physical object.
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For example, what if I turn the starfish
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by a sixth of a turn,
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and then a third of a turn?
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So I've given names. The capital letters, A, B, C, D, E, F,
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are the names for the rotations.
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B, for example, rotates the little yellow dot
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to the B on the starfish. And so on.
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So what if I do B, which is a sixth of a turn,
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followed by C, which is a third of a turn?
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Well let's do that. A sixth of a turn,
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followed by a third of a turn,
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the combined effect is as if I had just rotated it by half a turn in one go.
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So the little table here records
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how the algebra of these symmetries work.
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I do one followed by another, the answer is
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it's rotation D, half a turn.
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What I if I did it in the other order? Would it make any difference?
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Let's see. Let's do the third of the turn first, and then the sixth of a turn.
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Of course, it doesn't make any difference.
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It still ends up at half a turn.
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And there is some symmetry here in the way the symmetries interact with each other.
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But this is completely different to the symmetries of the triangle.
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Let's see what happens if we do two symmetries
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with the triangle, one after the other.
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Let's do a rotation by a third of a turn anticlockwise,
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and reflect in the line through X.
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Well, the combined effect is as if I had just done the reflection in the line through Z
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to start with.
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Now, let's do it in a different order.
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Let's do the reflection in X first,
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followed by the rotation by a third of a turn anticlockwise.
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The combined effect, the triangle ends up somewhere completely different.
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It's as if it was reflected in the line through Y.
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Now it matters what order you do the operations in.
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And this enables us to distinguish
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why the symmetries of these objects --
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they both have six symmetries. So why shouldn't we say
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they have the same symmetries?
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But the way the symmetries interact
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enable us -- we've now got a language
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to distinguish why these symmetries are fundamentally different.
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And you can try this when you go down to the pub, later on.
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Take a beer mat and rotate it by a quarter of a turn,
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then flip it. And then do it in the other order,
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and the picture will be facing in the opposite direction.
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Now, Galois produced some laws for how these tables -- how symmetries interact.
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It's almost like little Sudoku tables.
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You don't see any symmetry twice
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in any row or column.
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And, using those rules, he was able to say
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that there are in fact only two objects
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with six symmetries.
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And they'll be the same as the symmetries of the triangle,
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or the symmetries of the six-pointed starfish.
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I think this is an amazing development.
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It's almost like the concept of number being developed for symmetry.
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In the front here, I've got one, two, three people
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sitting on one, two, three chairs.
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The people and the chairs are very different,
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but the number, the abstract idea of the number, is the same.
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And we can see this now: we go back to the walls in the Alhambra.
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Here are two very different walls,
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very different geometric pictures.
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But, using the language of Galois,
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we can understand that the underlying abstract symmetries of these things
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are actually the same.
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For example, let's take this beautiful wall
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with the triangles with a little twist on them.
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You can rotate them by a sixth of a turn
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if you ignore the colors. We're not matching up the colors.
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But the shapes match up if I rotate by a sixth of a turn
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around the point where all the triangles meet.
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What about the center of a triangle? I can rotate
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by a third of a turn around the center of the triangle,
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and everything matches up.
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And then there is an interesting place halfway along an edge,
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where I can rotate by 180 degrees.
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And all the tiles match up again.
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So rotate along halfway along the edge, and they all match up.
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Now, let's move to the very different-looking wall in the Alhambra.
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And we find the same symmetries here, and the same interaction.
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So, there was a sixth of a turn. A third of a turn where the Z pieces meet.
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And the half a turn is halfway between the six pointed stars.
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And although these walls look very different,
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Galois has produced a language to say
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that in fact the symmetries underlying these are exactly the same.
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And it's a symmetry we call 6-3-2.
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Here is another example in the Alhambra.
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This is a wall, a ceiling, and a floor.
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They all look very different. But this language allows us to say
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that they are representations of the same symmetrical abstract object,
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which we call 4-4-2. Nothing to do with football,
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but because of the fact that there are two places where you can rotate
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by a quarter of a turn, and one by half a turn.
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Now, this power of the language is even more,
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because Galois can say,
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"Did the Moorish artists discover all of the possible symmetries
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on the walls in the Alhambra?"
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And it turns out they almost did.
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You can prove, using Galois' language,
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there are actually only 17
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different symmetries that you can do in the walls in the Alhambra.
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And they, if you try to produce a different wall with this 18th one,
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it will have to have the same symmetries as one of these 17.
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But these are things that we can see.
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And the power of Galois' mathematical language
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is it also allows us to create
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symmetrical objects in the unseen world,
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beyond the two-dimensional, three-dimensional,
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all the way through to the four- or five- or infinite-dimensional space.
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And that's where I work. I create
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mathematical objects, symmetrical objects,
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using Galois' language,
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in very high dimensional spaces.
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So I think it's a great example of things unseen,
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which the power of mathematical language allows you to create.
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So, like Galois, I stayed up all last night
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creating a new mathematical symmetrical object for you,
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and I've got a picture of it here.
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Well, unfortunately it isn't really a picture. If I could have my board
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at the side here, great, excellent.
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Here we are. Unfortunately, I can't show you
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a picture of this symmetrical object.
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But here is the language which describes
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how the symmetries interact.
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Now, this new symmetrical object
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does not have a name yet.
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Now, people like getting their names on things,
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on craters on the moon
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or new species of animals.
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So I'm going to give you the chance to get your name on a new symmetrical object
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which hasn't been named before.
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And this thing -- species die away,
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and moons kind of get hit by meteors and explode --
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but this mathematical object will live forever.
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It will make you immortal.
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In order to win this symmetrical object,
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what you have to do is to answer the question I asked you at the beginning.
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How many symmetries does a Rubik's Cube have?
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Okay, I'm going to sort you out.
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Rather than you all shouting out, I want you to count how many digits there are
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in that number. Okay?
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If you've got it as a factorial, you've got to expand the factorials.
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Okay, now if you want to play,
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I want you to stand up, okay?
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If you think you've got an estimate for how many digits,
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right -- we've already got one competitor here.
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If you all stay down he wins it automatically.
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Okay. Excellent. So we've got four here, five, six.
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Great. Excellent. That should get us going. All right.
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Anybody with five or less digits, you've got to sit down,
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because you've underestimated.
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Five or less digits. So, if you're in the tens of thousands you've got to sit down.
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60 digits or more, you've got to sit down.
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You've overestimated.
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20 digits or less, sit down.
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How many digits are there in your number?
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Two? So you should have sat down earlier.
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(Laughter)
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Let's have the other ones, who sat down during the 20, up again. Okay?
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If I told you 20 or less, stand up.
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Because this one. I think there were a few here.
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The people who just last sat down.
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Okay, how many digits do you have in your number?
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(Laughs)
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21. Okay good. How many do you have in yours?
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18. So it goes to this lady here.
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21 is the closest.
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It actually has -- the number of symmetries in the Rubik's cube
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has 25 digits.
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So now I need to name this object.
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So, what is your name?
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I need your surname. Symmetrical objects generally --
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spell it for me.
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G-H-E-Z
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No, SO2 has already been used, actually,
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in the mathematical language. So you can't have that one.
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So Ghez, there we go. That's your new symmetrical object.
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You are now immortal.
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(Applause)
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And if you'd like your own symmetrical object,
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I have a project raising money for a charity in Guatemala,
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where I will stay up all night and devise an object for you,
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for a donation to this charity to help kids get into education in Guatemala.
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And I think what drives me, as a mathematician,
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are those things which are not seen, the things that we haven't discovered.
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It's all the unanswered questions which make mathematics a living subject.
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And I will always come back to this quote from the Japanese "Essays in Idleness":
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"In everything, uniformity is undesirable.
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Leaving something incomplete makes it interesting,
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and gives one the feeling that there is room for growth." Thank you.
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(Applause)
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About this website

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