I would like to welcome you all to the Mathematics Institute. My name is.
I like O'Reilly, and I'm in charge of the public lecture series for the Institute.
Before we start the event, let me remind you that, as usual, the green sites indicate the emergency exit.
I also want to acknowledge a sponsor that you can see the markets there dispense of the whole Oxford Mathematics Lecture series.
X Market is a leading quantitative driven electronic market maker with offices in London, Singapore and New York.
And we're very grateful for the support.
Now onto the main event, as you can see or as you know, you can feel in the air, too, that tonight is a very special event.
It's actually co-organized with the Clay Mathematics Institute and Oxford Mathematics.
As such, it will be a little bit different from the usual format.
The public lecture today will be followed by the presentation of the Clay Award for Dissemination of Mathematical Knowledge.
I call it cat dunk for show. Good.
So you will have the distinct honour to be part of the event and the presentation, the award ceremony for the dunk.
And since we want it to be directly after the talk, we will not take question today.
We can go directly into the awards ceremony. Nick Woodhouse,
which is who is the former head of our department and the current president of the Clay Mathematics
Institute will give us a full description of the awards and of Professor Penrose's achievements.
So I will be very brief in my introduction for tonight's letter, I would just say the following When you work in mathematics and physics,
one sign of becoming famous is that your name may be associated with a theory.
We say, Oh, the theory, yes, that that person made it. Now you realise that Roger is in a completely different category when you see the partial list
of theories that has been named that either this name or that naturally associated with him,
for instance, you'll find a twist. The theory the notion of spin network cosmic censorship while curvature hypothesis
Schrodinger Newton equation andromeda paradox conformal cyclic cosmology just to name a few.
No, it gets even better when you name is use for a famous phenomena or theorem.
As you probably guessed, people don't usually name things after themself where some have tried, but it usually doesn't work.
So for that to happen, for a name to be given to you, a name to be given to something spectacular.
First of all, you have to done something quite remarkable.
But you also must have the admiration and the respect of the entire community.
And again, here, the list is quite long.
And I'm just naming a few people in mathematics and physics know about Penrose inequalities, Penrose interpretation of quantum mechanics.
The more Penrose pseudo inverse, the Newman Penrose formalism, Penrose diagrams, and of course the Penrose Hawking Singularity theorems.
Just to give you an idea. But things get even better and transcend science.
When you name are associated with things for the public at large, beyond physics and mathematics.
And here again, we have Penrose tiling that you might have seen coming in here tonight, the Penrose Cube that you can buy online.
Penrose stairs. Penrose Triangle. And of course, these last things are closely associated with this contact and influence on the artist M.C. Escher.
And this is a topic of tonight lecture. So without further ado, please let me welcome Roger and let's hear directly from him about this topic.
Well, thank you very much. Yeah, well, it's a great pleasure and an honour for me to be able to address this very distinguished audience here.
And I should explain the title.
First of all, it's a book which I'm supposed to be writing, and this is a way to get me started on this book.
I keep telling my publishers that I'm just one way getting through, and it's been nearly finished.
I haven't actually started yet, but this is the start in a certain sense because I had to organise my thoughts while I had done that much already,
but I had to organise my thoughts particularly I should explain.
The idea is to use pictures as an introduction to certain various areas of mathematics.
So it would be so, so that it would be a book for the general public.
And you would introduce the idea just by looking at your pictures.
I found it quite difficult in a way, because one of the problems is it's not as a sitting.
This actually illustrates this this one illustrates that. This illustrates the other thing.
You find that the same picture illustrates about three things all at once.
And then to try and put this into some kind of organisation I found extremely difficult.
This is something I can start with. Escher, I think when he was in Italy he did a lot of prints, lots of pictures in which he okay,
sometimes a little exaggerated but very accurate pictures of of scenery and so on buildings.
And I think it was when he visited the Alhambra in Spain and he realised all these
incredible Moorish patterns which he tried to understand what was going on.
There were different kinds of arrangements and then he learned from a mathematician, I think it was well, actually, I'm not quite sure where it was.
I did not say. But he got this picture, which just shows the 17 different Christian Christian groups, if you like,
symmetry groups of the plane where you have discrete motions and there are 717 different ones illustrated by these pictures you see here.
Well, Escher then developed these pictures in his own unique way, using tessellations of various creatures and so on.
I should explain. You see, he did this by.
In this one, you can see on the outside you've got the ordinary squares or something, and then he will modify them in terms of creatures.
This is the sort of thing you have. There are symmetries.
Either you have a translation which is just a motion.
Without any kind of rotation, you can have glide planes where you translate to note and you also reflect.
At the same time, you can have simple rotations.
That's the one down here. I don't remember. And all the other ones are.
Yes, the reflection. That's the other one.
So all these different motions combined together to give you the various 17 different groups that you can have.
And each one he would illustrate in his own unique way he was able to economise a little bit, although this one was to be in colour, I think.
But I just have a black and white version,
so you have to imagine they're all different colours and the two of these different groups are illustrated with the same picture.
If you take the colours into consideration, then you have one of them and if you don't, then you have the other one.
So, so you can do two at once. That's the sort of economy that he liked.
I shall just show these things rather quickly because I have an awful lot to show you.
There's another one and these groups are illustrated.
If you remember what those numbers and letters were and so on, and then you can see which one it is.
I'm not going to go through that in detail because I have too many other things to say.
But I'm just giving you examples of the way in which he illustrated these different symmetry groups, either with these birds and fish.
I said, look carefully to see which there are of them fish in that case, or we have in this case dogs.
And he liked the way that the I think they probably have it backwards doesn't make too much difference because this involves the glad plane you see.
So you have to turn it over to get the symmetry just as it is.
It's good, but you just can't because says okay upside down doesn't make too much difference in some the
particularly those ones and here we see another one in the groups is illustrated at the bottom.
I shall run through this rather quickly. Because I've got far more transparencies to show that I normally would have an election like this.
So I think you have to.
So is it not working? No, I suppose it's. It's just.
It's okay. That's okay. Thank you. And tried to see what was happening there.
Never mind. They can get very elaborate.
For example, with this one, you see the. The animals, the creatures.
I have to see what they are there and beetles or something of some sort.
And the arms all interlock and complicated way. But he was good at doing that.
And of course, you can different groups. You have to look carefully to see which group it is.
Did you here have a key at the bottom so you can tell. Anyway,
that that gives you a good way to start because he does illustrate these different symmetry groups
and one can go and talk about what groups are and motions of the plane and that sort of thing.
This one, this last picture here. Has a certain history to it, which I should explain.
I once met Asha. I come back to that story as to how I happened to get in touch with him.
But I did meet him once, and I came with a certain set of puzzle pieces, which in the shape of the ones that are at the top one minute.
I put my pointer down here at the very top.
I had a bunch of wooden pieces. They were I think they're wooden and they were the shape that we have here just.
And the thing was, you have to assemble them so that they cover the whole plane.
It was quite interesting because you have to have 12 different orientations, six one way up and six the other way up.
And they all have to be incorporated to cover the plane.
And then you can I had to he did it and wrote me a letter and said, well, he didn't quite see what the principle was.
And so I showed him this pattern here. Basically have run buses and these little lines.
Mark,
how you might have a this this one has to match the next one and you have to have a modification of the shape which so that one agrees with the other.
And this this is what it was. This one shape was designed so that a lot of false tries as well.
But that's that basically. And then I explained how it was done and I suggested some possible animal.
I think it was a bird of some kind, but rather crude. And I said, I'm sure you can do a lot better than that.
And he produced the picture, which I just showed you previously.
This is actually the only example he had of what you could call a non non Asahi.
Ideal tiling is it's a tiling of one shape, but the different instances of that shape may not be on equal footing with each other.
So although I think the dark ones and the light ones, I'm not sure if it is illustrated that way,
but some of them have a different relation to the passengers as a whole.
So you see two, one and another one. And if you move that one to the other one, the pattern doesn't follow it.
So you have that's what's called the non Asahi general tiling.
It's quite interesting historically because in the famous speech that Hilbert, when he introduced his his famous problems at the turn of the 19th,
20th century, he one of the problems he made was if you have a three dimensional shape,
is it possible that you have a shape which will tile only not so he really I'm not sure if that word was used at that time,
but that's what he was trying to do.
I think he thought the plane shapes are completely trivial and so there was nothing interesting you could do with plain shapes.
But then one of his colleagues or students, I think he I think it was he produced an example which showed that even in the case of plane shapes,
this conjecture was incorrect and that you did have shapes which would only tile the plane that would tell the plane,
but only in a way watch, which was non a hydrogen.
So that was an example. But anyway, the picture that Escher made with, I believe the last with certainly the last watercolour he ever made,
which partly due to my slowness in responding to his letters, but never mind about that.
Anyway, you can do other things with similar sorts of shapes.
Let me just give you an example here, very similar to some of the ones which did have periodic shapes.
But this isn't it's not obviously periodic. It may be it extends in some such way.
They're not all the same size. I'm not even quite sure what's involved in the arrangement here.
Of course, you can do more or less symmetrical things still by just having initials animals and things like that.
There's no symmetry at all. And of course, many of his examples or things like that.
What about three dimensions? Well, of course, that's a bit hard to illustrate, but he certainly did things in three dimensions.
Here's an example of of just cubical arrangement.
This one, I think this is what I wanted to show.
Here is a sketch of this picture.
That's another one, two of the three dimensional things, the things that he then his interest in.
And here we have another three dimensional picture, this fish.
And you can see that again, it illustrates a symmetry.
This is just translation. So it's not so exciting, but it's as a picture is impressive.
Of course, the only one I know of which which illustrates a tiling of three dimensions is this one.
Here you have these flatworms, whatever they are swimming around and the tiling that you see in the background,
this swimming through it and they just pass it of course is the tiling of regular tetrahedra and and octahedron and they fit together.
To form a three dimensional lattice tunnelling three space.
So you see that he was interested in three dimensions of his harder to draw pictures of those things so I don't know that
he did so many of the specific didn't do so many of them he also explored with distorting geometry in interesting ways.
And here you have an example where the building you go up and it twists around and you find a copy of the building up there.
But the geometry is twisted around in some strange way. And this is the sketch for it, I think.
So you see examples of that you rather like these creatures which he invented because they crawl along like a caterpillar or something,
and then they roll up and they can. When they go downhill, they can roll up and roll down the hill.
These are steps. Oh, that thing they do, it is they go out the door, you can see them anyway.
He rather like these creatures and made use of them in various ways.
But the the the geometry of the space is distorted in interesting ways, too.
So anyway, that's other things in three dimensions.
He like playing with polyhedra and also combinations of polyhedra such as here you have the two tetrahedra and there's a straightforward dodecahedron,
I think, and that's quite a team he played with.
I'll give you another example here, which is a.
I have to see what it is because I'm not sure I notice. See the picture.
It's a. You can probably tell me.
You can see it better from where you're sitting than I can. Anyway, these.
These are two polyhedra interlocking, which you have. I think it's an icosahedron, a dodecahedron.
It'll enter locking and preserving the same symmetry as each other.
And here you have, I think, three octahedron and lizard creatures and various things involved.
So and he's got other ones dotted around like that.
So he was quite keen on that idea of showing into penetrating polyhedra and the symmetry.
This is not the symmetry of the octahedron you have. When you finish that, you see you can.
I should have worked out what the symmetry was before I gave this talk.
But here we have then in.
I always like this one because you can see it's a planet of some sort and the two
planets are both tetrahedra and they interlock and they don't seem to know anything
about each other because one of these is all dinosaurs and things and the others
are some kind of civilised community and interlock in that extraordinary way.
So I'm going through this rather quickly because you really have to look at this pictures in detail to appreciate all the intricate,
the tremendous artistry has and detail which is extraordinary and it needs an examination.
So that one seems to be stuck to the bottom here. That one too. Here we have a this is just delighted.
That's the still isolated dodecahedron these.
A pentagram and passing through each other, and it has the same symmetry as the icosahedron.
But it's more interesting in that the faces go through each other and the polygons also go through each other as well.
And I think this is illustrating sort of various kinds of rubbish around here.
And then you have this perfect shape in the middle.
One of the things he liked doing was exploring the contrast between the image of a three dimensional object and the three dimensional object.
And here you see, this one looks as though it's a three dimensional object, and then one at the bottom is flat.
But nevertheless, it's a picture of a three dimensional object.
I mean, the spheres this one here is, I guess, a hemisphere or something.
But the top, you see a picture. So he's playing around with the picture of the object in the object itself.
And he does much more elaborate things of that nature.
And the striking one is this what you see in the background, these two, I don't know, humanoids or something.
We seem to be forming this interlocking pattern and then they sort of get a bit fed up and walk out of the picture and make friends with each other.
So that's one example. It has other examples of the same thing.
And this I've always liked that one.
You see the tiling here of these lizard creatures or whatever they were, and that could continue as a plain tiling.
But then they decide to come to life and they walk around on top of these polyhedra and other objects.
Then, after having had a stroke, they get back into the into the two dimensional picture.
It's a paradox, of course, but he loved that kind of thing. And then we have a much more elaborate version of the same kind of situation.
Also mirrors involved. So you can see the mirror, I guess the blind symmetry too, you see, because it's reflected.
You have a reflection in the symmetry. When they're black and white and somehow they go around and.
You just have to follow it. It's very, very complicated what's going on,
because they go through and they're also reflected and you see the reflections merging with the original creatures.
And here's another case where he illustrates two kinds of paradox at the same time.
I mean, one of them is this contrast between a picture of something and the thing itself.
Here you have a picture of a hand gets more and more realistic until you think it's a real hand, and that's drawing a picture.
And then, of course, it does the same thing there.
But there's also a kind of paradox in that the thing is the thing being drawn by this hand and therefore,
in some sense, this is a it's a product of this thing.
And then this is itself a product of that. So you have something which is is it's a bit like a Russell paradox where something is the asset
which has itself as a member and you see that kind of thing being illustrated in that picture.
I had this opportunity to visit that Escher, as I mentioned before,
and when I was there he had this long table I was expecting when I visited him that
he would have a very extraordinary house with staircases going over the window,
that sort of thing. And I was rather disappointed to find it was a very neatly organised house with a
lovely picture window of the view outside and a very neat and organised place.
And he had this long table, I don't know about as long as. Yeah, and it's about as wide as that.
And he was sitting at one end and he had two piles of prints and he said well look this size,
this one, I don't have many left but the other side have quite a few left.
You can choose one. So I thought, Wow. And so I brought the pile over and I started going through it.
And it was really difficult because some of the favourite ones he had already gone there were in the other pile,
but there were a lot of things and it was difficult to make the choice.
And eventually I picked this one, though he was very pleased when I picked that one because he said people don't usually understand that one.
And so he felt since I seemed to understand it, I thought at least I thought I did.
I was very flattered that he that he was pleased with my choice.
I should explain this a little bit. Here we have a fish.
And you see, as you move around here, the scales of the fish become a tessellation of other fish.
And as you go round, if I'm going which way around it is, and you find that one of these fish become this one here, and then it scales.
They have bigger and bigger and bigger, and then one of its scales become the original fish.
So it's a bit like that, the two hands drawing each other, but in a somewhat more subtle way.
And so it takes a little while to figure out exactly what's going on here.
But again, you have this sort of Russell Paradox idea.
You see the scales are a subset of the fish, but yet the fish, the scales become a fish and then it scales or one of the subsets,
one of the elements at least of this set of little fish, if you like, becomes the original one.
So you have a Russell Paradox idea involving sets,
which are members of themselves or members of other sets which are members of themselves, that sort of thing.
It's very much like a perhaps better known picture, the picture gallery where you see there's a boy.
I've got it the right way around. Yes, I think so. A boy over here somewhere who was in the picture gallery had to cover up the rest of it.
Really? To see that he's in a picture gallery and he's looking at the picture.
And then you follow the picture around and you see eventually that the boy is in in the picture.
So he's again, it's a sort of Russell paradox kind of thing where the boy is in the picture itself.
But it illustrates something else which was actually explored more by a mathematician, Lindstrom, who is a Dutch mathematician in London.
And he and a colleague decided, you see in the middle of the picture, Escher didn't quite know what to do.
Well, I don't know if that was it or what. He didn't quite know what to do with it.
He decides at least that was a place to put his signature. So it seems as though he gave up on trying to fill that thing in the middle.
He sort of filled it up in the middle with the other picture,
but it's done in a way which you don't quite see that there's anything strange about it, but I suppose it is a sort of singularity in the middle.
But here he gave up on the singularity and decided just to have a whole.
But what Lindstrom and his colleague decided to do was to find a transformation of this picture which would fit inside that hole.
And then, of course, there's a hole inside that picture. And so the whole picture goes into that as well.
And he made a very nice film of this where you sort of zoom in on it and you see the pictures inside itself all the time.
And it gives you a very eerie feeling. This just goes on and on and on with a slight rotation at the same time.
But to do this and I think Escher also used this sort of thing in it to use of bit of
complex analysis here because it's a conformal mixture and you use the conformal.
This is of course where one can talk about the mathematics in developing the idea of complex analysis and how the mapping of the complex plane,
polymorphic mapping to another part of a complex plane.
You always have these things which are these things which are conformal so that angles are preserved.
Sizes are not preserved. What was that?
Oh, I better start that again.
Yeah. So yeah, I tend to not my microphones off by waving my arms around but um,
so this is the sort of mathematics kind of lead into because it relates to the conformal
geometry that is closely certainly to the one dimensional complex analysis which is complex.
You have to have real parameters to describe your complex number. And so that's the normal, complex plane.
And then when you transform with a complex analytic function,
the or differentiable complex differentiable function, then the mapping is one which is conformal.
And the Dutch mathematicians don't make great deal of use of this.
In order to make this picture map into itself, you have to do something about making the square outside into a circle to make it fit.
But they managed to do this very well. Okay, well, conformal transformations are also interesting.
Escher made use of them here. I hope you can see this one.
It's a bit like the tessellations of a plane we had before, but as you work your way down, you see the sizes change.
And so it really isn't quite a Euclidean motion, but it's a little motion where you squash it down.
So it's one of these conformal motions which related to complex analysis and that would be very nice area to discuss, certainly in more detail.
One of the more famous of Escher's pictures is this one here.
And here we see one of his circle limits.
It's quite curious because it's very I find it very useful to use this picture to illustrate the different cosmologies in cosmology.
You usually make an assumption that the universe especially isotropic and homogeneous,
and these things were a limited number of geometries you can use, basically characterised by the curvature.
The curvature can be positive, there can be more complicated topologies,
but there are basically the three types the positive curve, zero curvature and the negative curvature.
And Escher was very ingenious in showing that he could use more or less the same shapes.
Illustrate each of these three. If you go carefully, you see the number of the devil's feet coming together as three.
Here was it, whereas it's only two there and there are different numbers of things that the the wings and so on.
And so you can get this without changing the shapes very much.
You can get these three different geometries illustrated.
This geometry illustrates various things, which I've often use it in the lectures on cosmology.
As I said before, it's an illustration of the.
But this is what's called hyperbolic geometry. That's the Euclidean geometry.
That's the geometry of the sphere. Hyperbolic geometry is negative curvature and is often eschewed in several of these as called circle limits.
He learned about it from Cox Sitter, who was at the same meeting of the International Congress of Mathematicians in 1950.
I can't remember exactly 53, 54 or something where I went to this in my second year as a graduate student,
and this is where I first became acquainted with Escher's work.
Escher didn't at that time have these examples here, but he let me go back to the one I just showed you.
You didn't have that then, but he learned from Cox.
Sitter two is attending this meeting and Cox is to explain to Escher.
He said, look, would be interesting if you could illustrate the hyperbolic plane.
This is the plane that people usually mathematicians usually refer to as the Poincaré disk.
I always worry about that because it was initially due to Beltrami and I guess people don't seem to remember it was Beltrami.
He had all these different transformations. He had the Kline representation.
They put incorrect half plane in the park or a disk before these other people did.
I'm not quite sure about Riemann because Raymond already had the half plane,
apparently, but I don't know whether he realised it was a non-Euclidean geometry.
I'm not sure. But anyway, this is this was the first one that Escher made.
It's actually quite useful to use that one because then you can see how this the you see these are the straight lines are the
straight lines diameters or else you have Oxford circles which make the boundary orthogonal and the since it's conformal,
the angles are correctly represented and you can see how you can have lines which never meet each other.
I mean, these would be parallels which which actually never meet had an ethical these parallels because
the parallel to infinity and then triangles you could see the angles don't add up to 180 degrees,
they add up to something less. And all that is well understood as a piece of mathematics.
But Escher had this very nice representation of it and you can see the way you you know, nothing figures if I get it right and.
Something about that anyway. And then this is a little bit about that.
As it does now, I've also used this where let's say this one as a nice illustration of a kind of geometry where infinity is represented.
You see the angels and devils actually, for that in the West is as well as this one, because you can see the fish here.
The black ones are supposed to be all congruent in the geometry. So you have to imagine that this fish and that fish that think they're the same.
The geometry is such that we represent it in this way, so they get squashed up towards the edge.
But as far as the fish are concerned, the white the white ones are the black ones because it is completely homogeneous.
This geometry that you see also in this example, the eyes of the fish are exact circles,
and these circles remain exact circles no matter how close you are to the boundary.
His line is extraordinarily accurately, right down to the edge. You can look at look at it with a microscope.
You can still see it's very, very precise, right down to the edge.
Now, you see this is useful. I find it useful in my discussions of cosmology because you can use the same trick for space times.
This is this is a two dimensional space. But in cosmology, you talk about space and time together and three dimensions of space, an amount of time.
And you could talk about conformal representations just the same way.
And so I find it very handy here.
We see a picture this time going up the page and this is meant to be, well, you see, can only represent two dimensions of space.
But I've got this wiggly part in the back because I don't want to prejudice the whether or not the space is actually closed or not.
It might be open or closed, so I've allowed for it to wiggle around at the back.
But to have it look as that starting to be closed is useful because otherwise you can't convey the the the size of the space very easily.
But anyway, there is the big bang here. The universe expands, slows down a bit,
and then starts to expand again with this exponential expansion which was recently observed and got the Nobel Prize and all that.
And you see, here's the trick. You squashed that down so it becomes a finite boundary.
This is a trick which has been useful to use talking about gravitational radiation.
And certainly I was in the in the sixties and seventies playing with that.
You can also do the opposite trick, which is to expand out the big bang here.
Escher. I was trying to find issue representations of this.
We did things rather like that. Here we have a picture as you guys to the middle and again,
it's conformal and they go right down to I think you have to cut a hole and stop
it for a bit because the big bang is really a surface rather than the point.
So you have to stop it off somewhere. But it is very another an Escher representation of this idea.
And here we have another illustration. So you could think of the big bang as somewhere in the middle there.
And it's control may represent it, but it's sort of opposite trick here.
You have infinity squashed on in there you have the big bang stretched out.
This is a perfectly respectable thing to do in cosmology.
What's not quite so respectable is this model, which is a model which which I've been playing with for a long time.
I think there's some evidence for it, which I won't go into here because that's not the right talk.
But that said, I just want to show you this picture that you can do something very Escher like, which is join the the big bang.
It's the stretching out of the big bang to the squashing down.
This is the this is our Ian, I should say big bang to remote future.
And then this is the next Ian after us, this is the one before ours.
And they form this nice continuous stretch with all these aeons going right back to infinity in both directions.
Okay, that's enough of that. I haven't played with the topology of the universe.
You can certainly have these homogeneous isotropic models and pull them up to various
ways to make several logical things that you did in a straight topology in various ways.
They were just some examples. See, this is a bit like a mobius strip that you have to go around four times I think before you get back to you started.
He did play with Mobius strips as well and you're going to see these ants or whatever they are
crawling over the Mobius strip and they can be one way or the other way on top of each other.
Nice things to examine. Okay, now let me talk about something else.
I did talk about tessellations of the plane.
And unfortunately, what I'm going to show you came just too late for Escher.
He died before I was able to.
I think I wasn't quick enough off the mark. If I say he died too early, I wasn't quick enough to mark this.
I want to show you this because it has some relation to the tiling that you see as you come in the building here.
And I think I want to explain how that idea is behind that tiling.
Well, this is the Italian I came across not long after she had actually died.
I think he would have appreciated this picture. I should explain.
It's a well-known theorem of crystallography, or, if you like, of the types of motions of the plane,
which I've been talking about, that the only symmetries you can have when you get translational symmetries.
But if you have a translational symmetry that's sliding along without any rotation together with a rotational symmetry,
then the only rotational symmetries you can have two fold, three fold, four fold and six fold.
And it's quite easy to prove that. But then here you have a pattern which seems to be very uniform.
It is indeed very uniform. I've expanding, but it seems to have a five fold symmetry.
You see, it's made out of pentagons, but actually four different shapes tend to consist of around pieces.
You have these tentacles, a five pointed star, and just as caps as I call them, sort of half stars.
It's quite interesting to look at even without knowing how it's built.
If you set your eye along where I'm standing better than where you are and you can see that things line up, every little line continues.
So with infinite number of line segments, whichever line you choose and the density of those line segments all the way along is the same.
You also have this interesting decadence, or the place you see regular decadence.
Regular decadence. Whenever you see a regular decadence like this one here where I live, I say it is not what it used to be.
So I have trouble finding these things. So let me try one by looking at it first and then come back to you.
Here's one there. The decorations are always the same.
Three pentagons, one just this cap and tour offices, wherever they are.
And whenever you find one of these decadence, it's surrounded by a ring of ten pentagons.
So I always rather like those rings. I'll come back to them in a minute.
The pattern is very uniform. In fact, although it never repeats itself, it can't.
That would violate the theorem. It almost repeats itself.
That is to say, if you give me, say, any percentage, less than 100%, say 99.9%, then I could give you a motion of translational motion of this,
such that the pattern agrees to 99.9% and also the rotation of the five fold rotation to 99.9%.
And then you say, well, maybe what, about 99.99%?
This is yeah, you could do that to any percentage, less than 100%.
You can find a motion which was sent this pattern unto itself, but it's never quite symmetric.
And that's how it wriggles through the theorem. Let me explain how this is constructed.
Here we have, I should say, is a sort of hierarchical arrangement.
And here we have a bigger version of the same thing.
Here we have a again, this is hierarchically arranged.
So now the bigger version of the same thing. And here we have another version of the same thing.
So that is the sort of organisation. It's quite useful to show that and then ask somebody, can you find these pentagons?
And it's not so easy, which is a way of illustrating where, well, how uniform it really is.
Okay. Now, this pattern, I think I'm going to use the second set of hierarchy.
This one here is a little easier to explain what I'm going to say.
Yeah, I can see the patterns more easily to here. Here you have the deck again and the rings.
I always like these rings, you see. I'll come back to that in a minute.
But it's. This is made out of four.
It shapes where the Pentagon is actually. Well, that's not great.
But there's another way of doing the same sort of thing.
These are cuts in DART and the relationship between the cuts in DART and the original pattern is this.
And you, if you look carefully enough, done it right.
You see each cut has the same set of lines in it and each as each other cut and each dart has the same set of lines.
And if you match the lines, then you get this arrangement here.
In fact, the only way of putting the lines together so that you match them is to make that pattern.
So these cuts in dart marks with the lines that you see here can only be assembled in a non periodic way.
And so this example was the first known example where you have two shapes which will only tile the plane in a non periodic way.
You can also Escher ises and this is the sort of things that came as she wasn't able to see this.
I'm sure he would have done wonderful things. I shall try and find the right spot here, which is difficult.
And I think to do it. That's not bad.
Yeah. I don't know how well I've done it, but you can see that this these are cats and dogs, but modified to look like birds.
So this is what I mean by ization. This is my poor attempt at trying to do something that Escher might have done as he lived long enough.
But other people have done other things like this too.
So it's interesting that you can do something which I'm sure you would've revelled in if you ever had the chance.
Right. Let me go back to the cats and dogs and so on.
What? To the initial. This one here. There's another way of doing it, which is these promises.
And again, they are related to the cuts and cuts with lines drawn on the promises.
And if you match the lines, then you get the original pattern back again.
And you can do it directly between cats, dogs and their bosses.
And it shows you can mark one of them to get to the other one and so on.
I want to try and describe the tiling outside the front office building because those are two dots, but what are the markings on those tiles?
Well, that came from the following, and I better make sure I don't get these things upside down when I pull them to the side.
Here we have. The original Pentagon pattern.
And as I say, I rather like these rings of Pentagon's. So it was nice to mark them.
We have these green things go round the through those rings at Pentagon's.
Now, that doesn't quite fill up the regions of the Pentagon.
So let's do the more to that. I'm going to add some more green lives.
And the virtue of doing this is that now I have a pattern which is really the same as that pattern, but it's a distortion of it.
So you see here we have the Pentagon's you've got three different versions of the Pentagon's.
They're very much distorted, but three different Pentagon's curvilinear Pentagon's now and you see the justice caps here is one.
And they put it this way, there's just this gap and there's just this justice gap and so on.
And so that is the same pattern, but somewhat distorted.
Now, can you get these onto a rhombus tiling where you can, but the trouble is that the rhombus isn't quite the same.
So you need to add a few more lines. Having edges, a few lines.
You now see that every fat rhombus has the same markings on it,
every thin rhombus has the same markings on it, and that is the tiling that we have outside the front.
So if you want to know what it is, that's what it is. It's, it's the rhombus tiling with all these green things on it.
So the markings there, I wondered for quite a long time whether the you have places where these rings close up and there are four
different ones where they close up and you can see all four of them with different patterns on the outside.
But there was. But are there any others? Well, the others would have to be ones which actually cross each other.
Cross them. And are there any which close up and cross each other?
I don't think there are. And the pattern there and I sort of conjectured that none of them did.
Realisation of that conjecture is completely wrong. I did find one which wiggles all over the place much too big for the pattern outside
the building here because all over the place and finally gets back to itself.
So that conjecture was wrong to have a new conjecture and I was that that almost all of them do close.
So you can certainly have patterns where where one or two of them don't close.
But if it's more than two, I think probably, you know, closer. That's a conjecture.
I don't know if it's true or not. Okay. That's that.
And this, in fact, is the pattern that I gave to people so that they would get it right.
Now, I want to give you something else. I've been playing around these things quite a bit, and there are other things you can do, too.
There is a set of three shapes, not just two, and they're quite simple shapes the square, regular hexagon and the regular dodecahedron.
And these markings on them are the things you have to match.
And I was interested to know if when you match them, what you see, do you get some feeling for the pattern as a whole?
I tried this with something before and I got no feeling whatsoever.
But Joshua Sokolow, I gave a talk when I mentioned this at the end, and Joshua Sokoloff,
the computer program, which was related to this, and you can see there is the pattern over there.
I'll show you the next picture, which is the next stage of the hierarchy.
And I was really quite surprised how it is really a very attractive pattern.
It's one of these things which never actually quite repeats.
It's not quite right in that those little wiggly, tiny rings ought to be exact circles, but that's the only thing that's not quite right about it.
They should be. The little tiny one should be exact circles. They look they actually look falling.
Falling. And.
Anyway, somebody perhaps could make a building and would have these three tiles and they can they'd need a lot of them to see that pattern there.
Okay, anyway, that's enough of this. Now, when I told you that I did go and see the an exhibition when I was in Amsterdam.
At the International Congress of Mathematicians. And it was in the Dunkirk Museum.
One of my lecturers, Sean Wiley, who taught me algebraic topology, and he had a picture of it on the catalogue which had the Escher picture.
Night and day I'm showing you that one,
with birds flying off in different directions and spaces between the birds picking the birds going the other way.
One of the tonight's scene of the day scene on the other side.
And so I went into this exhibition, and I remember I remember being particularly struck by this picture called relativity,
but all these different staircases and creatures going up the staircases of gravity being in different directions.
And I was really very struck by that. And, of course, many of the other things he had there.
So I went away trying to think I would try and do something impossible myself,
not with the skill that you had, of course, but at least with an idea that I hadn't actually seen.
And I produced things with bridges and rivers and that going off in different directions.
And I whittle it down to something which I thought illustrated the impossibility in its simplest sort of reduced it to its simplest form.
And I showed this picture to my father, and then he started producing staircases, buildings and all that sort of thing.
And we wrote a paper where we showed some of his pictures.
We didn't know what the subject was, but my father happened to know the editor of the British Journal of Psychology.
So we decided it was psychology. Probably they could get it published because he was good friends with the editor and indeed that's what happened.
So it was published and we then sent the copies to Esher, giving reference to his catalogue and saying appreciation of his work.
And in the meantime, Asher had actually produced something of a similar nature,
which is this picture of a dairy where you see the top part of the building is joined up in an impossible way to the bottom half.
He, after seeing the paper we had in the British Journal of Psychology, Ezra then produce his own version of The Staircase,
which is a very famous picture now that the monk's going around in both directions.
Very beautifully done. And Escher was very generous and giving credit to the paper.
This was the occasion when I said, I must meet Asher.
My father and Asher had a quite a correspondence backwards and forwards, and I had Escher's telephone number.
I was travelling in, in the Netherlands, I think in Appleton, near to Bonn, where I lived.
And I phoned him up and he said, Just come along. And he said, Come along for tea, which is what happened.
So that's how I got to know him.
But the triangle, which I'm going to show you, I'm going to say something about the mathematics that this is involved with.
But before getting to that, let me show you.
I was a psychology conference and I thought I'd try and go one step better than some of these pictures and produce an invisible, impossible object.
So this this is an invisible, impossible triangle.
This is also an invisible possible triangle and not quite sure which one is better.
And you can put the two together and maybe that works best.
I don't know. Anyway, you can probably see the invisible triangle.
It's not really there, but it's impossible, even if it's not there.
I was at a there was a film being made for some reason about trusses and things like that.
And the people it was shown, I think it was on the BBC, Montel a long time ago, and they were starting to make this film and talking about twisters.
And at one point they finally asked me, What's it good for you see?
So I said, Well, one thing you can use, Twist, is for solving Maxwell's equations.
And then I said, Well, well, you had this. You have to use a certain idea that I can't really explain it.
It's cosmology. I can't explain that to you, but it's. So they got very disappointed about that.
So then I went home and I thought, gosh, I can come on this course and they can explain it.
Here we are. You see, this is a nice way of producing that.
You see your mark. Imagine that you have some pieces which you have to assemble to make that object.
And these pieces consist of a lot of corners like that made they made out of wood, you say,
and you make these out of wood and you have instructions about how to build them together.
That's supposed to glue to that one, and this is to glue to this one in that glue to that one.
And you have to look at these instructions and see whether the instructions
allow you to build the objects or you're going to have a little problem with it.
And then I realise that actually is an element of cosmology.
If the cosmology element vanishes, you're just put and you see these instructions give you a check representation of an object,
and this object you're using the freedom you have of the distance from the eye.
And so that's what enables you to construct the current multi and nontrivial cosmology element.
And if the instructions have a non vanishing coloured cosmology element, then that's something you can't build.
If come on, the element does vanish, then you can build it.
So I thought this was a nice way into that notion of cosmology.
I see the measure.
It's a Czech representation of cosmology. I was talking about this too, at a conference in I think it was in Rome.
It was on that mathematics and asked and I can't remember something like that.
And I was talking to an American mathematician, unfortunately, regarding who it was.
And I was explaining that my talk the next day was going to be about cosmology and how this represents that.
And he said, Oh, well, is there any other group you can represent? I said, Well, I hadn't really thought about that.
So this is the sort of. You can take the distance from the eye and you take the logarithm of it, and that gives you a an additive group.
So you can then talk about the come on in the ordinary check way.
That has to be explained in more detail in the book, of course,
but then in the other groups I thought, well, yeah, there is, you know, I could think of that too.
And so, in fact, I drew this picture out and I gave it and I took the next day.
I think it's a little easier if I don't show you that one. But something which I developed a bit later.
It's the same idea. And what is this you see?
Well, this is an idea that you use. And let me show you this picture here.
It's the only one I know of his, which I'm not sure.
It's quite cosmology. I haven't quite figured it out yet that you see on one side of the picture.
I mean, Escher used this idea for other things where you have this sort of like a cube paradox.
The thing could be one way around or the other. You could see a cube edge on and you may see the back of it as well.
You don't know quite whether you see the cube internally, whether you're looking inside or outside.
And if you see this, you get it one way around and then that tells you the rest of the picture which way around it is.
So Escher has this here and this is clearly one way around and this is clearly the other way around.
But in the middle it's ambiguous so that the whole thing is impossible because you get the ambiguity in the middle,
which if you make one choice, which is this choice that contradicts that, and the other choice contradicts this.
But the thing I had here, I think as you would have done wonderful things with this, too.
But you see the staircase. You could imagine walking around the staircase and well, it's impossible.
I did show it to a but there's another version of it here.
I did show it to an expert on impossible objects who was at the conference,
and I showed him this picture and he said, Can you see what's wrong with this? It's nothing on that picture.
Could you make that out, would you think? Yeah, sure, if I could make that out. What are you really sure you could?
You have to work it because the trouble is your eyes, you go round it, it flips to the other one without you noticing.
That's what's happening. So you certainly couldn't make that out of a piece of wood anyway.
So that's that. Can you make these impossible objects?
Well, that's a good question. You could certainly make the pieces.
One thing you can do is this. In fact, this is a model of something which I believe somewhere was in college.
It's still there. I'm not quite sure where there is a bigger version of this.
I can't show it to you all at once, everywhere, because you see, this is the impossibility.
It's not a Czech version. This is more like a like aa1 of the continuous versions of Come On You Can have and that's one way of doing it.
But let me show you something else. Ah, no, that's the picture we just had.
Isn't this we've done that. Yes. Simple screen.
Wow. And wow, look, it still works.
That was lined up just right. So you have it's a possible impossible triangle.
And of course, if you want to see what it really is, you give us a little twist.
Well, first of all, I think the best expert for this we had.
There we are. Oh, we give us a little twist and then you can see what it is.
Okay. Well, thank you very much. Thank you very much for this wonderful lecture.
And now for the second part of the event, I'd like to please, I'd be welcoming a Nyquist house.
Who is going to take and present the awards, please.
Should I come back or. Okay.
So in a moment, I'm going to call on Lavinia Clay to present the Clay Award for the dissemination of mathematical knowledge to Roger Penrose.
So dissemination of mathematical knowledge is a very important part of the Clay Institute's mission,
alongside supporting mathematical research at the very highest level and of course, giving very large prizes.
The this award recognises the achievements of people who have made exceptional contributions under both headings.
In Roger's case, the award is in recognition of his outstanding contributions to geometry, relativity and other branches of mathematics,
and of his tireless work in explaining mathematical ideas to the public through popular books, public lectures and broadcasts.
But of course, Alan at the beginning gave a long list of his contributions within relativity, within geometry.
And that is takes very the first books very, very clearly.
But he is also, as you seen, a brilliant expositor of mathematical ideas.
And it's the combination of these two which leads to the award.
So we've seen his some of his contributions to geometry and the Penrose tiling.
But there's another aspect of his work which is really very important and which bridges the to the two, which is the passing from one area to another.
So he begins by thinking about something impossible.
So you start thinking about the idea of tiling the plane with Pentagons.
Well, most people might have occurred to earlier to think whether that's possible, and they decide very rapidly that it's not possible.
Well, Roger thinks about that. And what does it lead to?
It leads to the Penrose tiling. And that has an impact, a huge impact outside the area in which he thought about it.
And in that case, an impact in the visual arts and and beyond.
Or he starts thinking about impossible objects.
And where does that lead to? Leads to a paper in the British Journal of Psychology, which is there are other examples of this.
I mentioned briefly the more of Penrose inverse.
Well, again, you you learn at school about taking the inverse of a square matrix.
Well, what about a matrix that isn't square? Well, you give that a few seconds thought.
That's impossible. So you go on. Not if you're Roger. You think about it some more.
And that leads to this generalised inverse.
And again, that has a huge impact. In fact, his paper on the generalised inverse is his most cited paper.
It's a hugely useful tool in numerical analysis.
It's had a big impact on computation and other things.
Where he's crossed the boundary, perhaps is it was Roger who.
This one's for the computer scientist who first made the suggestion that the church's
lambda calculus might be a useful tool in thinking about programming languages.
And that, again, had a huge impact.
But he's best known outside the scientific community for his popular books, which are wide ranging, highly original,
highly controversial, in the very best sense of being controversial in that they provoke debate and get people talking.
So they have a very unusual feature for popular books in science in that they're
addressed not only to the general public in explaining mathematical ideas,
but also in engaging the public in the development ideas of ideas.
So he's he's putting forward his own scientific ideas, original ideas in his books in a way that many other people do not.
His books have been translated into many different languages.
They've sold in the hundreds of thousands, and they are a remarkable achievement.
Well, you may have noticed that Roger doesn't use the most up to date of the technology.
It is in his public lectures. But notwithstanding that, he does have a very significant online presence.
If you look. On YouTube, you will find some 30 expository videos, recording lectures and interviews.
Many of them are been viewed tens of thousands of times, or in one case, over 300,000 times.
So that's a huge impact there.
So I think it's clear that he very amply fulfils the criteria for our award,
both in the contributions that he's made in the development of mathematical
ideas and the incredible work he's done in the dissemination of mathematics.
So I'll now call on Lavinia Clay to present the award to him.