I. So why are we here tonight? They are, at least for good reason. So two years ago, we we moved in into beautiful building, the Andrew Wildes building. And the first reaction to many of us, many mathematician and mathematician, seeing all the arrangement of stairs, is this is just like Escher, painting is just like Escher. So we wanted to explore that a little bit because nowadays when people come in, they also they still say that the visitor says, Oh, that reminds me of Escher.
So Escher really resonates with many people, especially mathematician, as that vision of mathematics realised in print. Some of them get very obsessed by it, not me. And we wanted to, to, to push a little bit that inspiration of Escher in mathematics into, into an event. And the inspiration was stepfather with when John Chapman came for Sir John Chapman would be giving the second the second lectures came to me. So I decided to transform the building into the art gallery of Escher.
And we'll hear about the mathematics of that in, in about 30 minutes. Then the third things that happen is that Roger Penrose will here tonight. He was gifted by a series of different collector, private collectors for beautiful prints. And they're right outside. They're going to be outside just today. So I invite you when you leave. If you haven't seen them, you go right on the right after we're done.
And cities, original prints. So there are a series of six print press one, which is the waterfall. And we decide that we need to have a special event to unveil them. So you'd be pleased to know that these prints will be on permanent display once we find the right place in Oxford where to display them. And it will be it will make the Oxford University the largest permanent display of Escher in the UK and that by a factor of seven. So it will its prints are very special. I don't know.
It's it always takes a little bit of time to sort of things, so I don't know when they will be in permanent display. So but I invite you to to have a look at them tonight. The the last thing that happened is that inspired by this very event tonight, time, Hitchcock decided to make a movie about Escher and would be showing the Escher Escher movie. I know it sounds impossible, but tonight nothing is impossible. There is also, at the same time also inspired by all our work and events.
The exhibits are a temporary exhibition in Edinburgh, but all the Escher, a very large exhibition and TS has now moved to London. And it's today's the opening day. It turns up also. So I also invite you to have a look. It's very wonderful exhibition will scene in the movie. Yeah. So some of these some of these works. So all of that. I could not resist the temptation. We could not resist the temptation to organise an event related to Escher. And that's what bring us here tonight.
So the running event for tonight, let me tell you, with first wave, Clem Hitchcock was going to tell us about the movie. The movie is 28 minutes long. And right after that, John Chapman will tell us about the mathematics of Escher, one particular print and the one that he made following the same rules.
And then we hear from Sir Roger Penrose about his own dealing with Penrose, some of the actual impossible object that he managed to realise that we managed to assemble here tonight, but also the Escher from the future. What would Escher ever thought possible of made with modern mathematics with more, more, more new ideas.
So we are all looking forward to to do that. Now I see there are some young people here, so you probably don't know, but putting an event like that, it's an event like any other and like any other event that we've put together. And if you if you know about lawyers or insurance or administration, you realise that this is truly impossible to make an event like that.
I have I have original prints, original models and the panel that we have today, it is not a miracle, but it is the hard work of different people, in particular Darrow Lumber, that is that, you know, many of you know, Bella Sandro was in charge of the art. Some are with Son and Ruth Preston. So if I missed other people, but they really have made this possible. So as I as I welcome Trent for the introduction, please give them all of them a big round of applause. Thank you. Thanks so much.
It's great to be here. Thanks very much for having me on, particularly in this building, which features so very strongly in the film that you're about to see. And I would reiterate the but both this part to keep an eye out and the I wonder if the design is just new how you ask that stairs would look under under a lens of a camera. So it's fantastic to be here. I think with something like this.
It's great that a story like this to be told any story but a documentary like this, to have the enthusiasm of somebody. But also that connection to a particular subject like this, as Professor Penrose adds, is a rare thing indeed. And to get the chance to bring that out in a film like this is is wonderful.
I think that that a lot of stars have to align for something like this to be made in particular, as was just mentioned, the exhibition which remarkably, the exhibition that started in Scotland earlier this year was the first ever major retrospective of its type in this country. And it's amazing there's only one Escher work here other than that which is in the hunterian in Glasgow.
Amazing that now thanks to the collection at the Community Museum, so many are here and I would strongly recommend the, the exhibition which just moved to Dulwich and is going to be there till January, I think as well as the story that you're about to hear about the connection, professor's connection with Escher and Escher with mathematics. One of the things that was perhaps very surprising about the film for us was how overlooked Escher has being the technical side of his work has been.
And you can see in the works, outside works, it was so familiar with reproductions on prints and posters and on album covers actually to see the craft, the the woodcraft scale that's needed to make those images. I think that's something that really surprised us. I too have a couple of thank you's before I start. These are these wonderful models that you see, which are reconstructions of ones made by the professor and his father were made by Antony Penrose.
So I don't think he's here to put a hand that these were made, especially for us. And they're wonderful, as you see in the film. Also, these these dogs, which you see will complete the illusion in a way that will become clear soon, were were initially used for the models that the professor and his father originally constructed, and they were supplied by Shirley Hodgson, physicist as a thank you for that.
And finally, thank you very much, the professor for for telling your story in such a wonderful way that you're about to see here and for his patience in walking up all those stairs. Thanks very much indeed. Enjoy the film. Thank you. Thank you. Thank you. So the picture I want to tell you about. Find where to stand. It's this one which made a very brief appearance in the movie, but wasn't really discussed at all.
And it's English name is Print Gallery, and it's got some very interesting mathematics behind it, which was uncovered in about 21, 22 by a Dutch mathematician called Hendrik Leinster. And I left or came and gave us a talk in 2002 on it. And I want to share with you what I learned from him that day about the mathematics behind this print gallery. So what exactly is going on here? You have a guy in the bottom left who's looking at a picture, and in that picture there's a town by the sea.
And if you follow it around to the top right, then the buildings get bigger. And eventually you realise that the building that he's stood in is a building that's in the town that he's looking at so that he is standing in the picture that he is looking at. So you have this cyclic expansion as you go around. Everything gets slightly bigger until you get back. And of course, he is in the picture is looking at but so is a copy of the picture. And it carries on going around and around never ending.
What's interesting about this picture is if you look at any little bit of it, so I just choose the a little bit, then it looks perfectly normal. There's nothing unusual about it and it doesn't matter which bit you look at. It looks like a perfectly normal bit of a print. It's only when you put it all together that you get the illusion because this expansion is gradual as you go around.
So the first thing I want to tell you is how how I should generated this picture and then I want to tell you what the mathematicians did when they got hold of it. So you started with the idea of laying a grid of squares down on a piece of paper, and then what I need to do is to form the squared of squares so that as you move along, one of the boundaries, the squares get bigger. And the first thing you tried to do was to use straight lines to do this.
So you had this fan of lines coming out of the corner, so that if you start with them a certain distance apart here, by the time you get to this edge, they're four times as far apart. And if you do that, then the little square that you shade down in the bottom right becomes four times as big. By the time it gets to this corner.
And a little square that you shade here becomes four times as big so that by the time you've gone all the way round, you've increased your scale by a factor of four times, four times four times four, which is 256. But the problem with using straight lines is where you can see it in this picture. I started off with a little square and I've ended up with a shape in that corner, which is not a square anymore because the lines are not at right angles anymore.
And when you do this defamation and you don't keep the two lines at right angles, you end up sharing the picture and it distorts it. So that if you did look a little bit of this, it wouldn't look like a normal print. It would be too distorted. So I started this way and then threw it away and decided not to do this.
And intuitively he adopted a scheme where he used curved lines and the advantage of using curved lines was he could make sure that the original lines, which were at right angles to each other, would stay at right angles to each other everywhere. And the advantage of doing that is that you start off with little squares and you always end up with squares all the way around.
And he just found this this idea, I think intuitively in mathematics, we have a name for this sort of map where it preserves angles so that right angles stay right angles, and they're called conformal maps. And that was the conformal map of my title. So the actual map that you use looks something like this. So you can see that if you look at any little bit of this, it just looks like a grid of squares.
But the scale of the grid changes as you go around. So if I start with a little square and a by the time I get to D, that square is four times as big. And if you have a little square D and you did the same thing, got to see you, you'd get four times as big a gain. So by the time you've gone all the way around this picture, one little square would become 256 times as big. So that's the grit that he got. How then did you do you make the picture once you've got the grit?
Well. So you take your original unreformed picture, you do a sketch, and you lay this grid of squares over the top of it. And then what you have to do is that for every square up here, you look to see what's in that square, and you copy it out onto the same square, into the deformed picture. And when you do that and you do it assiduously for each one of these squares, where then that will automatically give you this expansion, the zooming in as you go around.
So that was the way that Asha generated his picture and led him to this by the time you filled in all the squares. So Leinster was fascinated by this and you asked himself two questions. So the first question was what did the original picture look like that actually started with before it was deformed? This is what the deformed one looks like. Can you reproduce the original rectified picture?
The one that would still have straight lines? And then the second question was, why did you leave a hole in the middle here? What should go in the hole? And if we know what the what the grid is, could we continue the picture into the hole? And what would happen if we did continue it into the hole? So I'm going to try and answer both those questions.
So the first thing that Lance did to try and get some information about about what exactly it was that Asher had done is we had this transform grid over here on the left, and then the original picture is somewhere here on the right. I don't know what it looks like yet, but let me assume the picture is sat behind here somewhere. And I'm going to walk around in this transform picture and see where I end up with in the original picture.
So I'm first going to do a path where I walk from A to B. So I've covered a certain number of squares in the transform picture and I will have gone a certain distance in the original picture from A to B, and then I'm going to turn left and go to see. Now, the squares on this side are four times as small as the squares on this side, because the squares were getting smaller all the way along here. So one big square here becomes a square, which is very small here.
So if I go from B to C, I have to cross for a distance, which is four times as great in the original picture, because the squares have got smaller. And if I come. From C to D, you have to go four times as far again because the squares are getting smaller all the time. So to go the same distance, you have to cross more squares. And then if I come from D to A, I've got to go 64, four times as far again.
So in the transform picture, I've now got back to where I started from, but in the original picture I haven't got back to where I started from. I've got a lot further out. And in fact, I'm a factor of 256 further away from the centre of that picture than where I started from, because I got this factor of four for every time I walked along with one of these edges. So given that in the original picture, I'm back where I in the transform picture, I'm back where I started.
That means in the original picture, whatever is at this position must be the same as it's at this position. I must have a copy of it so that I got back to where I started. And that tells you that whatever this original picture is, it's invariant by scaling by 256. So that if you took this picture and you blew the whole thing up by a factor of 256, it would look the same. So this is known more or less universally now is the Droste effect.
After a Dutch brand of cocoa powder. So so this is a very famous cocoa powder in Holland. And you can see that this is a picture of the box with a maid and she's carrying a tray on, which is the box. So she's carrying the tray. That is the box. That is her again. And of course, on that box, there's a picture of her carrying the box. And on that box, that's a picture of her carrying the box and so on, ad infinitum.
So this picture carries on. It gets very small because of the factor of about five each time. But it's invariant in the sense that if you blew this box up so that it was the size of this box, then that little box would be this box and etc., etc., and you would picture would look the same again. And I can give you a live demonstration of this if this camera is working. This is the sort of thing that it's called video feedback. So is you.
But if I pointed at the screen then, now you see the picture of the screen on which is a picture of the screen is a picture of the screen. And it carries on all the way down. So whatever the original picture was, it has this invariance by a factor of 256. Okay, let's do another walk. So I'm going to start again from a at this time I'm going to walk.
I'm going to go eight squares up. So in the original picture, I'm going up by distance eight and I've counted eight squares up in the grid, and now I'm going to turn left and I'm going to work eight squares again. The squares are getting smaller, so I don't get quite so far. I'm going to work eight squares down and I'm going to turn left again and walk eight squares to the right. So now in the original picture, I've got back to where I started, but in the deformed picture I haven't.
I've got to some point here and the same argument works again now. So because the original picture I must have the same thing a day, then must be the same thing at this point as there is at this point in that transform picture. And that immediately answers the question as to what should go in the middle of the whole. That this is this point here is effectively in the middle of the whole. So what they should be there is exactly the same thing that you see.
But shrink and I have to rotate it a little bit because there's been a bit of rotation here as well. So the transform picture as well is invariant under scaling and rotation. So in that little hole in the middle, there should be another copy of the whole thing shrunk down and rotated. And of course, in that copy, there is a little hole in the middle of that in which there'd be another copy shrunk down and rotated and so on all the way in.
So I'll show you a demonstration later on where I have a version of this picture where we don't take the whole out to show you what really goes in the middle. Okay. So I'm going to demonstrate the map in a different way now. But 256 is too big a scale factor to show you anything that if I had a if I shrunk this box by a factor of 256, you wouldn't be to see anything. So I'm going to use a factor of four instead. So here's a grid which has the drastic effects, but with a factor of four.
So if I shrunk this outer grid down by a factor of four, it would be the same as this grid and shrink that by a factor of four. It's in the middle of grid, etc. You have to imagine this going all the way down, but I've only drawn three levels of it. So if I blew this picture up by a factor of four, it would look the same because this grid would become that one and the middle one would become this one. And then the one I hadn't drawn in the middle would come and fill out the inner one.
And just to show you so you can keep track of which grid is which I'm going to colour code them. So what you have to imagine is that if I draw a picture on this, whatever, I draw on the green, I draw the same thing on the blue, but scaled by a factor of four. And I draw the same thing again on the red but scale by a factor of four. So this picture carries on all the way down. And then I want to show you what Asha's map does.
It cuts this picture along that the dark line that you can see there, and it shifts one side relative to the other. One is getting slightly bigger, ones getting slightly smaller, and then you stop when it exactly matches up again. And so now you wouldn't see any cut because even though the green is not joined up to the green anymore, it's joined up to the blue. And the blue is a perfect copy of the green.
And the size has been shifted so that it exactly matches so that instead of this thing now being invariant as you go in, I have this you can sort of see the spiral expansion now. But as you go around, things get smaller, but it joins up perfectly. And you can also see on this picture that the red is a copy of the blue, which is a copy of the green.
So you can see, because I've only got a factor of four, you can see that this picture repeats itself still as I go in, but the green doesn't join up to green anymore. Now green joins up to blue. And that's why as you go around, you stood in your own picture as opposed to having a new picture in the middle. Okay. So that's another way of looking at this map. But I still haven't told you how Leinster worked out what this map was.
So how would you actually work out mathematically how I generated this picture? How did I know what map to use? So to do that, Leinster divided the map into three stages. So the first stage is to take a logarithm. So let me show you what a logarithm does to this picture. It opens it up along that cut to gain. But this time it opens it out and folds it back. And you get something like this. So the. Let me explain what the act is.
So if in the original picture I can describe any point in this picture by using two numbers, one is the distance from the centre and one is a bearing or the angle. And when you take the logarithm of that picture, what you find is that on this axis you get the angle. And on this axis, you get the logarithm of the distance. So the three things to notice now about this picture.
The first is that this map that I just did is conformal, so that the lines that meet at right angles still meet at right angles. So little squares are still little squares. The second is that when I do this angle, I've only shown you the angle going between, well, -182 plus 180 here. But of course, you can carry on around as many times as you want in this. So if you carried on and went round a full loop, this picture would just repeat itself because you'd get back to where you started.
So really, this picture goes on, carries on, going upwards there, but it's periodic. It has a translation invariance. So you get this bit that I've drawn would appear again above and above that, likewise below and below that. So it has this. You can slide it and it looks the same. And that the reason for taking a log is that this now does the same thing in the other direction as well. So with this original picture, if you multiply by four, you get back to where you started.
But once you take a log, this the one formula that I have in my talk is that the logarithm of four times R is the same as the log of four plus the log of R. So by taking a log, you turn a scale invariance into a translational invariance. So in this picture, multiplying by four corresponds to adding log for to the log R. And so that's why the blue is a shift of the red and the green is a shift of the blue so that this picture is now periodic in this direction,
but it's also periodic in this direction. And that's the first step. Okay. That's the hard step. The second step is to rotate this picture. So I've added a little bit more on the top and the bottom. So you can see it would carry on periodically and now I'm going to rotate it and scale it a little bit. And I'm going to stop when the joint between the blue and the green is exactly above the joint between the blue and the red.
And by doing that, this picture is still periodic in the direction with the same period of 360 degrees. Because the blue is a copy of the red and the green is a copy of the blue. So I've got the same invariants. It's still got the same translation invariants in the vertical direction. And the fact that this line exactly maps onto this line means that I can undo the logarithm and the picture will join up smoothly. So when you undo it, so it stitches it back together again.
But now instead of blue joining on onto blue and green. Joining on to green, because I did that rotation. Now green joins onto blue and blue joins on to red. And that's how you make this map. So the key thing was, was getting that rotation right in the logarithm plane. And there's only one way to do that. And once you know what that is, you can work out what the formula is for this map. So let me show you.
Now that we know the formula, we can apply the inverse of this map to this painting and see what the rectified area would look like. And this is what it looks like. So this is the map without any distortion. And you can see that everything looks perfectly normal. But you might ask. Well, well. So first of all, what's this bit missing? That's the bit that fell outside the boundaries of this painting.
So we don't know what should be that didn't draw it. But you might also ask, well, where's the town gone? Where's the ship gone? Where's the rest of it? And of course, I've only shown you this at one scale. This is invariant under scaling by 256. So I've got to zoom in a bit to the middle to show you what the rest of it is. So if I zoom in by a factor of four, you get this and you start to see the ship now and you zoom in by a factor of four again.
You start to see the rest of the town and you zoom in by a factor of four again. You start to see the outside of the gallery, and then you zoom in by a factor of four again and you get back to here. There's another guy and you can keep going round. It's invariant under zooming in by 256.
So when Leinster came and gave us a colloquium on this in 2002, I was one of these people who was fascinated and obsessed by it and I thought, how cool would it be to actually do one of these, to actually take some pictures that have the right property and then apply the map? And it took a little while before technology caught up with my aspirations. But when we finally moved into the new building, I had the opportunity.
The computers could now do it. And I had a nice picture of a new building that had nice red grid structures that was going to look nice. So we went off and took some pictures. So I updated it a little bit instead of having a guy in a print gallery. I had a guy at a desk looking at a computer screen. And then Alan and I went to Green College Town next door, and we took this nice picture of the Mathematical Institute.
And the reason it's hard to do this is that one picture is not good enough, because if you do one picture and you zoom in by 256, you just lose all your resolution. Even if it's a 16 megapixel picture, it just doesn't work. So you need to take multiple pictures and they need to really fit together nicely. So you zoom in on this one. My office is just there.
So you zoom in on the middle of that and you get to this picture and then you zoom in on that and you zooming in just on this office and you see that. Then you can see there's a little bit of Photoshop going on here because when you take a picture of a building from the outside, you just see windows look like that. You don't see people like that in the middle.
So you have to do a little bit of Photoshop and then you zoom in on this and you see the guy at the computer screen and then you zoom in on this again and you get back to that. So you have it. It's cyclical again. And in fact, now that I've got pictures and a computer, I can really show you what this dress effect looks like by turning this into a little movie where you zoom in. And it keeps going, as in this guy. And of course, it'll keep going for as long as I show it. Okay.
So I've got my four pictures so I could a whole lot of each one. And then the idea is that so this picture fits in that whole half the scaling and this picture fits exactly in that whole after scaling and this one and that whole and then of course, this one and that all carries on all the way down. So now if I apply the math to each of these pictures, I get these four bits.
So even though each of those pictures was a factor of four smaller than the others, so the relative to the original picture, there are very small part of it. Each one contributes an equal amount to the final image. And so if I've got those exactly right, these should all line up when you put the four bits together. It clicks and you get the actor image, which is outside.
And our building really lends itself to this picture because the nice straight lines you see, the flowing curves that assured also wanted to do in his picture by making the print gallery with a row of prints. So you have a guy looking at a computer screen, and the building he's looking at on the computer screen is the same as the building that it sat in. And we blew this up and there's a picture of this outside the door.
You might have seen it on your way in, but you can certainly have a look on your way out. Now, so I said for my picture, I haven't taken the middle out. So in the middle there, the scale of 256, there's another version of me sat there at a different computer screen and I can show you that by zooming in again and rotating. And as you zoom in, you get the same thing back again. This.
Okay. In the last two or 3 minutes, I just want to show you some of the other things you can do once you know what this map is and some of the things that Escher might have played with if it's still been around. So let me just go back to how we generated this map. We we had this periodic structure. And so let me just put a grid around it. And I'm going to put. Put these dots at the edges of that structure.
So this is one period. So these are like the tiling. So each one of these squares is repeated. So whatever's in this square gets repeated in every other square. And then the way I generated the map was to rotate this and scale it so that this point light exactly above this point. So I chose this red point and I turned it into a point that lies exactly above the one that was fixed. But the map would work if I chose any other point here.
There's no special reason to choose this one. So I could choose any point here. Rotate in scale and put it at the position of this red one and the picture would join up. It might look strange, but it would join up. So I just wanted to show you some examples of what you get if you choose other points. And I just put them all on the same picture. So this is the original one where it's the one I just showed you. This one is also interesting.
So if you if you rotate, you choose this point, you know, rotate clockwise instead of anticlockwise, you also get a sensible Escher style picture. It's just that now it expands as you go anticlockwise and gets smaller as you go clockwise. But it's also something that that you could equally have drawn. It's rather unfortunate because of the scale factor as you go up the side. So this guy has got a very big head. This guy's got a very little head on which one is better.
And various other ones have these. You see that they get they get rather strange. This one has got it expands twice as you go around. So you have an even stronger factor. And most of them the direction of gravity doesn't stay vertical as you go around the outside, which is why they don't look so good. But maybe that's not such. It's something that we're in actually too much. And some of his other prints. Okay.
And the last thing I wanted to say was this number 256 that's come up all the way through. Right. That there was a factor of 256 invariants in the picture you scale by 256, you get the same thing again. So was there anything special about 256 Y 256 and now that I have these images, I can change what the scale factor is just by changing the size of the image I'm looking at. So if I zoom in at the picture on the screen, I'm changing the scale factor in the drastic effect.
And you see 256, you don't really notice really it's a Droste effect till you get close to the end where you get to this is a factor of four and now suddenly you can see multiple heads here. So you see that you are looking at a picture that has this video feedback, this Droste effect. So from here I can show you what the map looks like if I want to ask this. Didn't do very smoothly. We'll see if the computer can do that again. Yeah. So you shift the head along. Right.
You choose one head and you're shifted along and it joins on to the next body. And when you've done that now, it's fairly obvious that you have this spiral going in there and that inside there's a smaller version. And then inside again, there's a smaller version all the way down. So this picture also looks interesting, but it looks very different to the original. So a factor of four is not very good for generating that original illusion.
I think sort of central to this illusion was that you didn't know that it carried on infinitely, often into the middle. You didn't know what was there. You only saw one version. It's here. You clearly see multiple versions. And now I can scale back out again so I can show you what the actual picture would look like at various different scales. So. Let me see if I can pause it a few times. You scale out. Gradually the guy in the middle gets smaller.
You start to see more of the picture. You've got to scale out a long way before you're looking at something interesting. But all of them work. They would have a bigger white dot in the middle. And then you finally get to the original one. So maybe there's nothing special about 256 per say, but it needs to be a reasonably big number to get the effect that Asha was looking for.
It was too small. Then you would really I think you'd have a big gap in the middle, or you would see multiple copies of the image and it would be a different type of picture. Okay. I'm going to hand over to Roger. I'm just going to leave you with if you're interested in looking at any things. There's a paper that Leinster and De Smet wrote on the work that they did that's available. They have a nice website where they have lots of images of the maps that they that I've shown you.
And then some of the pictures that showed how you generated this in the first place are from the book by Bruno Ernst. Okay. Well, it's a great pleasure to be able to express my appreciation of Escher and to try and tell you a few things which perhaps he would have done things with if he'd lived longer. Well, that, of course, is another one. But I want to start by talking about the tiling outside the main building here. You saw that in the film and you saw it as you came in the door.
And that is the kind of thing that Escher might have made use of. And I would like to show you one or two ways, and then I want to go on to something else and we'll talk about these models over there. Let's see. This works. You may well have seen this picture. It's just one example of an Escher, but it's one of the ones outside the door. And it's an example illustrates the symmetry.
So you can see that the whole picture, if you move this dog into this dog by sliding it along, you can also turn it over and move it up as well. It's what's called a glide symmetry. That's one example. Now I'll show you another example which you saw in the film. Now, this one has the property that it has rotational symmetry. Now you see that it has points. Well, here's a point of twofold symmetry.
So you rotate that one so you can rotate through 180 degrees and the whole picture goes into itself. And here's a point of sixfold symmetry, and then you have points of three for symmetry. So if I can find one in the middle here somewhere. There. I think there's one. So he exhibited some of the crystal symmetries that, if possible, here. Now, when I say crystal symmetry, I mean something which can be associated with a translation of symmetries.
And those symmetries are exactly as illustrated here. We have the two fold symmetry that we just saw and the three fold and the six fold, but also four fold. So I should say that that's four fold and they're six fold. Now, I want to show you something else. Now you can see this picture is made up out of Pentagons, five pointed stars we called pentacles rhombus. That's an endowment shape. And this half star just as cap over here. It just uses this for shapes. And this thing created the pattern.
Well, I think I'll circumvent this legal problem by showing a different version of it. So you can see the pattern, the larger area of the pattern, and it's made up of just of those shapes. If you just look at it casually, it has a look as though it is has a translational symmetry. It looks almost as though it's a crystal pattern. However, and here's why I would need to show you the whole thing, because it looks like a Pentagon on the outside, and in fact, it has a five fold symmetry.
And that's not one of the crystal symmetries. Now, that's a theorem that you can only have those ones I told you. And I'm not going to explain how this gets around the theorem. But it does it by being almost five fold symmetric in a very important technical sense. Now, I want to show you how this pattern is constructed. So let's go back to my previous one. I hope I can get enough of it on to give you the idea. Now you see that these pentagons fit themselves together to make bigger, big tickets.
So let's take a look at one. And the picture is a Pentagon there, and that's subdivided into six smaller ones with these little triangular gaps. And that happens all over the place. So you can see that the Pentagon's in this picture can be collected together to make bigger pentagons. And so there we have a bigger pattern. I think I'll leave this one here because I want to take another stage. There we have the next stage. Those orange ones are again collected together to form these big new ones.
And the blue ones are not here. I have a problem because. I won't get that all that good. But you will see as you follow around, there is a big blue flag of him and the whole pattern is constructed in this hierarchical way. Now, the hierarchy is not a thing that you sort of notice. Look at the pattern. It looks very irregular and you don't really see any hierarchy. You see a regularity about it, which is very strong.
In fact, there are lots of features, such as if you take a little line, I'm not sure I can do this at the scale, but here, take one of these lines and you follow that line all the way along and it keeps on going. Any line in the picture, whichever direction it's pointing, it, it just keeps on going. So I guess it's hard for me to watch that and this at the same time, but moreover, it has interesting sub patterns.
Now, whether I can find one here, you see, here's one where you see a deck again, a ten sided figure, and these deck against all over the place. And wherever you find one of these regular ten sided figure, you'll find it's got three pentagons, two rhombus, and one just as cap. And they're always like that everywhere. Whenever you find one of those is always surrounded by a ring of ten pentagons.
So this one's got a ring of ten pentagons. And you've got these rings, which are rather fascinating when you look at it, and they sort of stand out. So there's a lot of order in this thing, which is nothing to do really with the hierarchical arrangement. It just has this kind of local order, which is which is highly symmetrical and which in a sense shouldn't be because it's one of forbidden crystal symmetries.
Now, I don't know where I can show you this, but I can I can show you them one at a time. Here I have you see, this is just a mathematical way in which you can construct this picture. But I had the idea is maybe you can force that arrangement with a kind of jigsaw puzzle. So here we have six pieces. Well, there are three pentagons. That one. That one. And that one. And we have these pentacle, the Justice Cup and the and the rhombus.
They are decorated with these little knobs and things. So that's how you make it into a jigsaw puzzle. You have to fit the little knobs in the right holes and so on. Six. What the [INAUDIBLE] are you doing? Excellent. That does help. Thank you.
Okay. I'll just tell you, I'm not going to show you why, but if you assemble these six pieces, if you imagine an infinite supply of these pieces, then they will fit together only in the kind of way I've shown you, which will be almost periodic, but never quite. So it's almost a crystal, but not quite. It's the sort of thing people call quasicrystals.
And actually they now find actual substances which which have this kind of symmetry and the atomic arrangement seem to follow the kind of thing that I have been showing you. So I had this is a sort of curious jigsaw or mathematical puzzle where you've got the six shapes, which will only fit together in a way which never repeats itself.
And I was talking to a mathematician, Simon Coachman, from Princeton, United States, and he was telling me about another set of such shapes, Robinson, which had been produced in 1971, based on some earlier ideas from how long and one of his students, and that he had reduced these original considerations to the six set of tiles. And I was told that, well, he was somebody who likes to get the smallest number and he got the six and nothing like mine, as you see.
But they have the same property that they will tile the whole plane, but only in a way which never repeats itself. That's what's called a periodic set of tiles. And but when he told me that Rafael Robinson likes to get the smallest number, he has said, well, I know I can do five, because if I take mine and then, you know, just this little spiky thing down there, the only place it can fit in is in here.
And this one's got one of the spiky things and the only place it can fit in here, it's all I need to do is cut out this piece, stick it on there and stick it on there. And then I've got five pieces. So then I went home and I thought, Well, I wonder if I can do any better than that. And I eventually came up with two.
Now, I remember feeling disappointed that you might think the strange reaction be disappointed because I thought it is so simple that surely it's well known and I just want to relate it to the original set. You see it take a little while to see this, but every one of these two shapes, which John Conway called cuts, that's the cut there. And dots, cuts of dots of every dot has the same pattern inside it and so does every cut.
And if you fit the match them together in this way, then you can match the cuts. And what's that where you get this? Now, in fact, you can force the matching by making the colouring, the vertices, either white or black. And if you want to match the white and black, the only way you can fit them together, the two now is in this never repeating pattern. So that's done with two. So we've been talking about issue here.
So I did wonder at one stations that should really be influencing me with his intriguing all sorts of ways of making periodic patterns into into interesting animals and so on. So I wondered, can you assure this? So I came up with one. This is not my first attempt. I'll show you that in a minute, but it's almost my first attempt. And you see, it's made up out of two bird shapes. There is the movie The Fat Bird, if I can find one.
There's the fat bird there. And the thin bird, which is this one here. And the only way that they will fit together is in this never repeating pattern. And if you want to know what the relationship between this is and the cuts and does it really is cuts and does in disguise is what we do. I hope I have lined this up right. We have to get it like that. And I think you will see that the little birds, the darts and the big birds are the cuts.
Well, the bird count is a bird, too, but I don't mean that. And so here we have the issue recession that I came up with of this type of pattern. Now, it's actually been around this time, unfortunately, just a few years. But sometimes my father did and neither of them saw these things, which was a great disappointment to me. But nevertheless, I have some interest. Now, there's another way of doing the two tails, and in some ways even simpler.
That's where these rumba shapes again, you must have matching rules. Of course, there are lots of ways of fitting. We're on this list together. You could just take one of them and just do it periodically. So you've got to have a rule which stops you doing that. Stuff, you've got to match it so that the spots is exactly the same spots that I had previously. And in fact it is exactly the same as what I showed you previously, but just rearranged a little bit.
So that is the same tiling, but things cut apart a bit and reassembled. And so there we have the rhombus version with the matching rules, and the only way of assembling that is to run the shapes with this set of matching rules is in a periodic or non periodic way, which is called a periodic set. Okay. Well, when I heard about this wonderful building as it was being constructed, Nick Woodhouse was one of the.
Well, he was the driving force behind all this and suggested to me that it might be nice to have a a tile at the front based on one of my tiling. So I thought, well, look. Lots of people have used this usually in that way without the decorations and seemed to me maybe we could do something a little bit more interesting. And if you remember going back to the version I have here, there were these rather nice rings, these decorations.
And every time you have a deck, again, it's surrounded by a ring of ten pentagons. So let's think about those rings and I'll mark the rings. Actually, it's better if I go to the next stage of the hierarchy. So. So let's go to the orange ones there. And I mark those rings of just around the deck decorations. And you see here we have a a green ring which goes round that deck again and it follows through the set of ten pentagons. And that happens all the way over. So you can put those rings down.
Oh, that would make a nice pattern because it brings out this rather attractive feature of those rings. Of course, just doing that, we have a lot of gaps all over the place. So in order to kill that little problem I'm going to put to make sure I've got it the right way up, which I think we have. Yes, there were a few more lines on this. So you see, now that I have a sort of curvilinear version of what I had before, the the the here's the pentacle.
You see, if you follow the green lines around, it's a sort of curved version of it. And the rhombus as a curved version of the Rhombus and the Justice Cup, there's a curved version of it. And then the Pentagon's get distorted in various different ways. But the three different versions of the Pentagon is distorted in different ways anyway. So there you are. It makes a nice pattern, but what does that do when you go to the to the rhombus? It just disappeared on top of it.
Go to the rhombus is instead to take away the Pentagon's because that confuses restricting but on business there and now we have a way of marking the embassies. However, we still have a lot of gaps. Some of them are different, so we better fill those gaps in, too. And when I've done that, I can find the right way around this thing. Yeah. Now every fat rhombus has the same crossed green lines on it. The everything about rhombus has two arcs on it, and they fit together to make this pattern.
And this is exactly the pattern you find outside the building, except that it isn't green, but you have this very nice thick stainless steel box, and that is the pattern that you walk across when you come into a building. So if you want to know what it is, that's what it is. Yes. Okay. Now, let me say something more about Sharon's issues. In fact, I think I'm going to go to the other device now. I could thank you, except it hasn't done it.
Oh, yeah, no, that's not the first one. Oh, can I go to the first one. Yeah. This is actually the first version Escher ization I did which appeared anywhere in the article I had. And you can see that it's the cuts in the dots really, but slightly distorted to make two birds. Now I want to show you another one. So let me move this on. Various people have tried to do this, and the person who's I think done the most in the most interesting ways is Richard Hassell.
And this, as far as I know, one of his very early examples, the little frogs, and it's the sort of thing Escher might have done. Might well have done. The bullfrogs are actually catching ducks. And I think I'm going to have to come back to this screen. If you do it for me, it'll save me from ruin everything. So here we are. Good. Thank you. This is the version of the Richard Hassel picture. And just to show you that, it usually just cuts dots.
Here we go. And I think if I've got it in the right place, you'll see that the big frogs, they're very are really got a very distorted you see there, their arms and legs go way out in again. But where they all come together with the arms and the leg come together are the vertices of this picture. And this big white one here is a kite. This one is a cut. That brown one is cut. It's dark brown. One is a dart here. And you see this white one is a dart and so on.
The colour coding I think is just a three colour coding through colouring of the entire pattern, and that's just one way you can do it. So I'm sure wish you would have done more things other people have done others. I think this is one of the nicest ones I've seen. I don't want to show you a whole lot because I want to go to other things now. In fact, that was just the tilings that, you know, say something about. Oh, before I go onto sorry, I going to show it's not just another picture.
This is a I just want to show you that the five fold one is just one of several. This is a 12 fold one. It's really very attractive due to go and then listen. There's four fold ones. So I should say eightfold ones, 12 fold ones and other well, seven fold ones look rather horrible. But the but the the eights and the 12, I think are very beautiful, particularly the twelves. And I've never seen a translation of this pattern, but it would be very nice to see that somebody has a go at it.
Okay. Now I want to move on to something else that's very important doing, which is the impossible objects. Now, you see, you couldn't make that out of wood, but we're going to see in a minute down here that you can do things like this. Is it impossible or not? No. I guess we won't be doing the camera doing. I hope this is going to work, because it really depends on nobody having juggled this table.
Nobody having job of this table. There was a lecture in here and then we had other lectures and wow, look, it still works. That was lined up just right. So you have 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 it a little twist and then you can see what it is. And I think that's a very nice model made by my cousin Tony.
And my father made one of these a long time ago. I don't know where it's got to. I think it may be in the basement, in the science museum somewhere, but it's very it was much harder to make it to get it out than to make a new one. So. So, Tony, this really not very nicely done on this with very little indication of what it actually looks like, I think. Okay. So that's that one. No, I think I can take this away. And it's the stepchild next, isn't it? Yes. No. Why don't I take that off? Yes.
No, it's tricky. This one. Arm it so that. That. It's just on top of that one. Just a little bit. That's it. Nearly. Really? Yeah. They just about I don't know how to direct, you know, line up the black strips I guess that better. That's it. And then slide it up. Excellent. Very good. They are. Thank you. Yes. Just.
My father made a model just like this, which was in the article that we wrote in the British Journal of Psychology, which we sent to Asher and which stimulated him to make his own very, very remarkable picture with the monks going round and round. Now, I want to say something about these dogs now. Now, how do we get. Can we get that thing to work? So this was something about these pictures was something that intrigued us. When you use perspective, here we go.
Yes. When you use perspective, there is a feature which she never actually used. And here you can see it lined up. I think we've got it just right here. You see the dogs? Where should we start? The front here. You see, you have what looks like a little dog and a big dog. But then when you follow around, you see the dogs and they follow rather all the same size as each other and they all come round. And then this one just just behind that one there.
So it looks and they look as though they're the same size. And these look as though they're the same size as the others. But when they get to the front, you see they're not. And there's no break in the front at all. And whereas the break where you see the range is, oops, what's happened? All right. All right, then. Now we know where the break is. One going to. Anybody who's inside there is it? So we took it. Oh, you've got a bat? Yes.
You see, the idea is the break is disguised by this little fellow having his back legs glued onto this piece here. So it looks at, though, front legs on this part over here where they're not just hanging into space. So the poor dog would fall over if he didn't have his loose back legs moved. But you see, that perspective is locally completely correct. That is to say, if you were to draw, it's like the previous picture we were saying this is locally completely correct and consistent.
It's just that globally it doesn't make sense or it makes a certain kind of sense. But it's not something you could construct out of wood without having a brake somewhere or bending it or something like that. So here, because that dog is in fact a lot further away from the eye from that dog there. You can have this dog physically bigger than this one, and they look as though they're comparable size.
So as you follow all the way around, it looks as though the sizes haven't jumped until you get here, which is where there is the jump. You can see quite clearly there's no break in the staircase. So it's a very nice illusion. And I'm sure Escher would have done things if it brought the perspective angle in which for some reason he never actually did. I want to show you an example. If I now go back to the back to the power point, we can do that.
I'm not sure if it's the next picture, but it should be. That's right. Yes, actually, that's. Yeah. Let me let me do a bit of history first. This is a picture which was made by Oscar Reuters Fahd, who is a Swedish architect. Well, I'm worried about architect artist, let's say Swedish artist who actually drew a picture like this in about the year I was born. So certainly things like this, it wasn't exactly the trend because it's a lot of squares, but it's the same idea.
So this is a thing which was done by Oscar Isaac, right? His father. And he did a lot of other things, including various staircases and things like that. So it's worth, I think, pointing out that ideas were explored earlier. Now, do I do it on this machine? No, I don't. Oh, here's the gadget over here. That's it. Is that right? Oh, this is just a send. Same thing, but done more slick.
You see, that was his, I think his original sketch. And then you can make it look much slicker with modern technology. But let me show you something else. You see that wasn't by any means the first example of an impossible object. Yeah, we have a very beautiful picture which was done by Pieter Bruegel, the elder, I suppose, in the 16th century.
And you see in the middle it's the name of the picture is the, is the magpie and the gallops, and there's a magpie sitting on the top of the gallows, the gallows in the middle of the picture.
And you see, it is an impossible structure because of where it's joined up at the top is different from the way the to I I think this point to is if we have these two places where the thing standing on the rock and they are sort of side by side and that's inconsistent with the way the thing is joined up at the top.
So it is an example of an impossible object. And if you go way back, I think there are some even earlier ones, but I think this is a wonderful one to show that there are examples of this sort of thing in art and you have to look for them. I think people look at this and say, a Brueghel made a mistake in this picture. Well, that's, of course, ridiculous. He knew what he was doing. And I think he wants to create some kind of eerie feeling about it, which you couldn't quite put your finger on.
I think that was the sort of he was trying to do. Now, this is a picture that I drew in an attempt to bring out this paradox that it didn't make use of, but could have. About the size. It's the dog paradox. You see the dogs. If you try to do it with perspective, then you have this issue of the sizes as you go round. The sizes are not consistent. So I was deliberately doing that here in a very extreme form.
So over on the left, you see this little child who's playing with this little train set just about to push it over, I think. But then there's little, little toy creatures walking up there, and then they climb up the staircase and they become the same size as the children playing at the top. So there's this inconsistency arising from this size problem. The size illusion comes about when you combine perspective with the impossible structure.
Now, I was at a conference in Rome, I think it was in honour of Escher and I happened to be talking to a mathematician. Unfortunately, I've forgotten who it was, but I've talked to a mathematician and I was pointing out that these things are illustrations of what mathematicians call comb ology. And I'm not going to explain what cosmology is, but roughly speaking, it's the idea that you can have something with a local structure which has a certain ambiguity.
And the ambiguity in the picture is you don't know how far away the object is that you're looking at. So you might be looking at a very small thing, close up or a much larger thing a long way away. And that freedom in the picture is something you can't get rid of. But the fact that you've got that freedom, you can use it in this inconsistent way. And that inconsistency, the measure of this inconsistency is what mathematicians call homology.
And this American mathematicians. What can you do that from the other group you say? And he said, Well, what about Z two or Z two? I guess he would have said and I thought for a bit, well no the thought of that before, but then when I thought about it a bit more, I thought, yes, you can. And so I wanted to use something that Escher used here. We had this picture before in the film, but he used the very striking effect.
I think you can see best what I mean if you hold your hand up and cover vertically half of the picture. Now if you look at the left hand picture, you have a certain interpretation, you've got a chap climbing up stairs and then you go through that thing and then there's somebody kind of dozing, kneeling down, and then there's a pool there, and then you try and go up the other stairs and you start getting into problems.
Now you just start from the other end, cover up the left hand part with your hand, if you like, and you see a perfect consistent picture. And this chap is climbing up the stairs and then he goes inside. And this thing is a kind of a well covering of a lamp or something like that, and it's completely different. You see, it's, it's, it's, it's the other way round. But what Escher has done is he's used the ambiguity of which way around it is, it's called concave and convex.
And which way around is it? You see, there's this freedom. It could be one way or the other. And on one side of. Any one of those is consistent on the other side. The other is consistent, but you have to have the strip in the middle where it's ambiguous. And that ambiguity ambiguity allows you to to do this impossible thing here. But I thought that there was another thing you could do, which is in some ways a bit more subtle. Although my picture has no comparison, of course, with that one.
But let me just show you. This was a picture I came up with. I had them on a simple one at a conference, which I showed the next day. Now, this was an example of something which illustrates the comb ology. Now, you see, I don't quite know how to describe this because it flips one way or the other a bit too easily. But if you try to be consistent. Suppose on the right hand side, that shaded part is the floor.
Then you can walk up the stairs and you can go to the left and walk upstairs, then walk upstairs. Welcome, sir. Then walk onto the top and then walk down stairs. And then down here. And then down here. And then down. And then you find that this is the floor, the white one, and it's flipped. You see, the flip is. Is a global thing. You noticed I probably need that trick that we had in the previous talk to feel what went on in the middle.
I didn't have an idea of how to do that, so I just put a little design in there because you do have a problem. Maybe you can think of a way of doing that. But at the moment I wasn't about what would the middle I mean, I had that far too. So I suppose I can get off of that. But you see, you do have an inconsistency. Inconsistency in the picture. There's somebody who's a real expert called Bruno Ernst, who actually dug up, I think, the rival picture, which I show and various things like that.
And I showed him this picture and I said anything wrong, you know? Sure. Could you could you make that out of wood so you're sure follow your way all the way around and it flips. Took him a long time to realise that that's an impossible object. And so there's a certain subtlety in that, which I suppose the, the convex and concave aspect of it flips too easily. And when you're walking your way around it jumps to the other and you don't realise it's done that.
But if you absolutely consistently all go all the way around, you find that you have an inconsistency. So it is an impossible object of a different kind, which again, I suppose Ashleigh would have done great things with and he could have. Well, there's another one. You see, this one has five. You see, you have to have an odd number going around making a system or a number, which isn't a multiple of three, I should say. And then this one is another one which is using seven now.
And the shading was always a problem. You have to be very careful about how you do the shading that that that works. Right. I think I will end up by showing you that the sort of thing that Escher presumably would have done if he had this kind of idea before him, and I'm sure he would have. This is this is an issue, a well-known Escher picture, and you will see is involves actually some combination of the restricts.
But the main thing I want to say here is if you follow one of those strips around, you see the little bubbles point one way or the other way. But if you follow them consistently, they're inconsistent with what the other ones do when you in fact, when they when they are at the side or the top, you see, when the poking out, you can see which way the bubbles have the point. But when you when they're sort of in the middle, they could point one way or the other and he's done it. So it's inconsistent.
If they point out on one side, you follow them around, then they shouldn't point out on the other side, they should be pointing in there. And it's a very clever you see is it's you can the ones I showed you we're just using straightforward flat flat surfaces were more or less flat and they're just sort of staircase things. Whereas here you have a much greater subtlety in the shading on these little bubble things.
And Escher would have, I'm sure, made of great use of that sort of thing to create impossible structures of kinds that he never have seen and never will see, except that I suppose other people can use these ideas to produce impossible structures. Thank you very much.