This is actually a chocolate box that Escher designed for a manufacturer in Holland that was celebrating an anniversary. When I retire I want one of these boxes. They are of absolute ___ [0:45:09]. If you go to the Escher Museum in Den Haag they've got an example. I presume they made many of these. They're little tin boxes but they're the most beautiful things.

For the symmetries of these tin box are actually one of the first non-obvious atomic symmetries that you can have. Because, actually, how many symmetries does this object have? Well, actually it's the same. It's the symmetries of a football. So the football -- it is a football made out of pentagons and hexagons but actually the symmetries of these are the same as the symmetries of the icosahedron [0:45:46].

How many different moves are there? Now, I can't reflect because Escher's done his clever twist. So he's twisted the starfish so there isn't a reflection, right? My six pointed starfish. How many different moves are there that I can make? I can do that. That's -- so I just got to keep the red things all aligned. You find that when you do this, you can do 60 different moves, 60 different symmetries of this shape. Now come on, 60. That's an incredibly divisible number. The reason the Babylonians used it for the base of their number system because it's so divisible. It's the reason we have 60 minutes in the hour. But when you consider the symmetries of this object, it turns out although the number 60 is very divisible, actually the symmetries of this can't be divided into smaller symmetrical objects. You might say, well, hold on. I mean, there's a pentagon here. Surely, the symmetries are the pentagon. That's a symmetry which is smaller hiding inside here. But if I try and divide by the pentagon and trying to get a symmetrical object, it doesn't make any sense. It turns out the mathematics that Galois [0:46:49] developed that there's no way to break this down into smaller symmetrical objects. This turns out to be one of the first interesting atoms in the periodic table of symmetry.

Galois [0:47:01] discovered this one but for 150 years, as Galois's [0:47:05] work developed, we were trying to pacify more and more these symmetrical objects, trying to find the atoms which make up symmetry. It culminated in the 1980s with what we believe was the completion of our periodic table. It included the most extraordinary objects including this amazing object which is a -- I mean, I can't show it to you because it's an object, it's a symmetrical object which leaves 196,883 dimensional space. Even in the last ___ [0:47:34] we can't make that object for you physically. But amazingly, it's a bit -- what do you want to think about it is this kind of like a weird snowflake which exists in this high-dimensional space and it's symmetries -- it's got more symmetries than there are atoms in the sun. Yet, we've discovered that we couldn't break this object down into smaller symmetries. Weirdly, this object just didn't have any part [0:47:58] it's just sort of weird objects that suddenly appeared in this high-dimensional space.

But how about the mathematicians [0:48:05]? Let's pull back a little bit. What do I mean by 196,883 dimensional objects? I mean, how do mathematicians play around with objects in such high dimensions?

Well, actually, this is another example of how fantastic language is in mathematics. I mean, language in a way, allowed Galois [0:48:24] to take the physical world of symmetry into a sort of Aunge Brackwell [0:48:27]. Suddenly, the pictures disappeared. In fact, there's a similar trick that Descartes made which allows us to see mathematically shapes in high dimensions, something we all learned in school, Cartesian geometry.

Cartesian geometry was basically a wonderful dictionary, a language which translates the world of shapes into the world of numbers. Say for example, if I want to describe a square in numbers, I can -- I'll locate it's corners, east, west, north, south location. A square consists of a point at the origin [0,0] and that's like one set along the horizontal axis, [1,0]. One set up the vertical axis is [0,1] and then a step [1,1] both horizontally and vertically. In a way, I can translate a square into just a sequence of numbers, pairs of numbers, four pairs of numbers identified for you as square.

If I want to go up a dimension, a cube, well, I can draw you a cube but I can similarly use the dictionary to translate it into numbers. It then becomes eight, triples of numbers which correspond to how many moves are you making in each direction, from north, north, north. 1 north, north, north. North 1, north, north, north 1, all the way through to the extreme point, [1, 1, 1].

What about a four dimensional cube? Now the point is this dictionary is wonderfully powerful. Because although the visual side runs out off to three dimensions, I can't show you a four dimensional cube, still the numerical side, the numbers side, carries on. I can tell you in numbers what a four dimensional cube is. It's a shape which has points at the Cartesians, [0,0,0,0] and then from that point you got an edge going out to a point at [1,0,0,0], [0,1,0,0], [0,0,1,0] and [0,0,0,1]. It's a four edges and then I'll just keep on going making steps in each of these directions until I get to the edge points at [1,1,1,1].

How many corners has a four dimensional cube got? 16, exactly. You were able to work out how many different ways there are putting 0s and 1s inside in there, 16. You can also use the same trick to work out how many edges there are. Now you may now move to [0:50:45] explore very quickly using this language the geometry of a shape you cannot see. A little thought should be able to get how many edges there are there and that's the amazing thing. It's that we can also start to explore the symmetries of this four-dimensional cube by using the same coordinates of things.

I can actually show you a shadow of this shape. In fact, if you go back to Paris, there is a shadow of a four-dimensional cube actually up at La Défense [0:51:14], the financial area in Paris. The arch at La Défense is a shadow of a four-dimensional cube in three dimensions. If you think about a three-dimensional cube, actually if an artist is drawing a three-dimensional cube on a two-dimensional canvas,

what they do is to make a square and then they put a small square inside and then they join up the edges. Suddenly, you got a sense of depth. You sort of think you're seeing a cube. Of course, it's just a flat thing.

Here's the same concept. If you project a four-dimensional cube into three dimensions, what you get is a small cube inside a larger cube and the sides all joined up. In fact, I have an example of it. Here's a four-dimensional cube projected down into our three-dimensional universe. You can see, you correctly articulated that there are 16 corners and you can see them, the [0:52:02] bodies little white balls and actually, you'll be able to count all of the edges now. Dare I do the math. There are 12 in each cube on there. So that's 24, 25, 26, 27, 28, 29, 30, 31, 32, so 32 edges. But actually you could have done that without this picture, but if I -- trying to articulate what an edge is in terms of these numbers.

If you're talking about 196,883 dimensions, shadows aren't going to be much good for you. Trying to explore the symmetries of this shape, you really do have to use the mathematical language which translates geometry into numbers. Amazingly, they bought us search and navigate [0:52:43], I'm a bit -- even for most mathematicians is pretty an amazing feat to be able to explore the symmetries of an object out in this number of dimensions.

One of the masters of the world of symmetry who helped to complete this classification of all of these symmetries is Professor John Conway in Princeton. He's one of the masters of symmetry. In fact, when I finished my degree here in Oxford, I mean I've fallen in love with the world of symmetry. It was about 1985 and I just heard about this completion of this project. In fact, Cambridge seemed to be the place they're going to study symmetry. In fact, they've produced this amazing book, it's the -- there's a picture up here but here's the real thing, the ATLAS of Finite Groups. This basically inside here are all the building blocks. It's the periodic table of symmetry. It was produced in about 1985. I just went on finishing my degree here. I went up to Cambridge to talk to John Conway and his group to try and find out what are next steps after having done this.

So I was breaking to join their group and he sat me down and he said, "Well, we're all very obsessed with symmetry here at Cambridge. So what's your name?" I said, "Okay. Marcus du Sautoy" And he said, "Well, you have to drop the 'du'." I said, "Why? Do you have something against French Aristocracy or something?" He says, "No. The authors in this, all [0:54:14], only have six letters in each of their names, Conway, Curtis, Norton, and Parker and Wilson." So Sautoy is fine, S-A-U-T-O-Y but not the 'du.' So I had to drop the 'du.' I was quite keen to join their group so I was very prepared. "Marcus, you've got any other initials." I said, "Yeah, NPF." "You're going to have to drop one of those letters." I said, "What?" "Yeah, we only have two initials in us, JH, RT, SP, RA." I said, "Okay. MP Sautoy, yeah." [0:54:43] That's not all, in fact. The first to join the group was John Conway and then he got his Ph.D. in Sweden and turned [0:54:51] Rob Curtis to join and there was this guy hanging around in Cambridge. He just kept

running around the office. He seemed to be quite helpful called Simon Norton, so he was the third to join the group. Parker joined, he was very good at computing and things. He joined. The last to join was Wilson. They all joined in alphabetical order. So if I was going to join the group, I was going to have to change my name, not the Sautoy but Zautoy would have been all right. So rather, by that point I realized, well okay, I love symmetry but this is going a little bit too far. So I was a little bit concerned though. I came back to Oxford wanting to start my research into the world of symmetry and I wonder, well, maybe symmetry is finished. Maybe well understand everything there is to know about symmetry.

That turns out not to be true. It's never true in mathematics. When you saw one problem, it opens up so many more interesting challenges. In particular, this thing called the Monster, this strange snowflake in 196,883 dimensions, it's very curious. Why should something suddenly emerge in this dimensional space? Has this number got any significance? It turns out it does. This number you'll find in a completely different area of mathematics called Number Theory. This number comes up into something called modular forms which is very much related to the work with Andrew Wiles did on Fermat's Last Theorem.

If you calculate the other dimensions where suddenly this Monster appears, all the numbers to the dimensions have something to do to this completely different area of mathematics. One of the challenges has been, we made some progress but I still think there's a lot to unlock, why has this strange symmetrical object got anything to do with the world of Number Theory? In fact, this is called Monstrous Moonshine because John Conway thinks that there's somehow a third object which is shining light on both of these other things [0:56:43] which the moonlight we're seeing is this strange numbers that are appearing.

In fact, Richard Borcherds won the Fields Medal and Nobel Prize for making a lot of deep discoveries about what -- the sun might be that shining the light on that. But Conway still believes, that says, a lot we still do not understand about this strange, symmetrical object which is one of the strange things in this ATLAS of Finite Symbol Groups.

But actually my own research -- I'm interested in, if this is the periodic table, then what are the molecules of symmetry that you can make? What can you build? What are the new symmetries that you can make out of this list in this periodic table? I spend my time here at Oxford, doing research in symmetry, trying to build new symmetries out of the building blocks that we've got in the ATLAS of Finite Symbol Groups. I discovered strange new symmetrical objects with connections to things called elliptic curves. In fact, I've got one of these new symmetrical objects which hasn't got a name yet. This is going to price for you, that I'm going to name this symmetrical object off to the person who can work out how many symmetries the Rubik's cube has. I even got a little certificate here. It's not for the name yet. The Somethink (ph) [0:58:04] group, this could be your name.

Science, you see, science is a process of evolution. One theory gets overturned by another. The science the Ancient Greeks proved, it looks far schooled from our modern perspective. But the amazing thing about mathematics is it's eternal. This group will be there forever. Species will die, stars will blow up but this group may give you a little bit of immortality if you can claim it. The challenge is, how many symmetries do this Rubik's cube has? Okay.

So I put it back there, the Rubik's cube here. How else I’m going to do this? Okay. All I want you to do is if you've got an estimate and you can make a guess, I want you to make a guess as to how many digits this number has. If you calculated the number and want to make a guess, how many digits does your -- so do you think it's got a hundred symmetries so it's got three digits, okay?

If you want to play, could you please stand up? I'm going to try and sort out. So make your estimate, okay? Does it got three symmetries, or the number of digits. Well, excellent. We've got some players. All you need to do is guess and you get a bit of immortality if you get close to this, okay? You only have to estimate. I don't need a precise formula. I just want you to estimate how many digits you think there are in this number. Okay. I'm going to sort you out a little bit. You're still calculating away, look at that and very impressive.

If you got ten or fewer digits, I want you to sit down because you've underestimated the number. It's got more than ten digits. Okay, good. That sorted a few of you out. If you've got 30 or more digits, you also must sit down because you've overestimated, okay?