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I. So why are we here tonight?

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They are, at least for good reason. So two years ago, we we moved in into beautiful building, the Andrew Wildes building.

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And the first reaction to many of us, many mathematician and mathematician,

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seeing all the arrangement of stairs, is this is just like Escher, painting is just like Escher.

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So we wanted to explore that a little bit because nowadays when people come in,

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they also they still say that the visitor says, Oh, that reminds me of Escher.

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So Escher really resonates with many people, especially mathematician, as that vision of mathematics realised in print.

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Some of them get very obsessed by it, not me.

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And we wanted to, to, to push a little bit that inspiration of Escher in mathematics into, into an event.

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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.

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So I decided to transform the building into the art gallery of Escher.

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And we'll hear about the mathematics of that in, in about 30 minutes.

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Then the third things that happen is that Roger Penrose will here tonight.

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He was gifted by a series of different collector, private collectors for beautiful prints.

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And they're right outside. They're going to be outside just today.

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So I invite you when you leave. If you haven't seen them, you go right on the right after we're done.

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And cities, original prints. So there are a series of six print press one, which is the waterfall.

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And we decide that we need to have a special event to unveil them.

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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.

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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.

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So it will its prints are very special. I don't know.

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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.

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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,

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Hitchcock decided to make a movie about Escher and would be showing the Escher Escher movie.

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I know it sounds impossible, but tonight nothing is impossible.

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There is also, at the same time also inspired by all our work and events.

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The exhibits are a temporary exhibition in Edinburgh, but all the Escher, a very large exhibition and TS has now moved to London.

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And it's today's the opening day. It turns up also. So I also invite you to have a look.

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It's very wonderful exhibition will scene in the movie.

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Yeah. So some of these some of these works. So all of that.

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I could not resist the temptation. We could not resist the temptation to organise an event related to Escher.

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And that's what bring us here tonight.

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So the running event for tonight, let me tell you, with first wave, Clem Hitchcock was going to tell us about the movie.

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The movie is 28 minutes long. And right after that, John Chapman will tell us about the mathematics of Escher,

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one particular print and the one that he made following the same rules.

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And then we hear from Sir Roger Penrose about his own dealing with Penrose,

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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.

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What would Escher ever thought possible of made with modern mathematics with more, more, more new ideas.

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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,

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it's an event like any other and like any other event that we've put together.

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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.

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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,

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in particular Darrow Lumber, that is that, you know, many of you know, Bella Sandro was in charge of the art.

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Some are with Son and Ruth Preston. So if I missed other people, but they really have made this possible.

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So as I as I welcome Trent for the introduction, please give them all of them a big round of applause.

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Thank you. Thanks so much.

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It's great to be here.

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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.

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And I would reiterate the but both this part to keep an eye out and the I wonder if the

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design is just new how you ask that stairs would look under under a lens of a camera.

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So it's fantastic to be here. I think with something like this.

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It's great that a story like this to be told any story but a documentary like this, to have the enthusiasm of somebody.

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But also that connection to a particular subject like this, as Professor Penrose adds, is a rare thing indeed.

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And to get the chance to bring that out in a film like this is is wonderful.

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I think that that a lot of stars have to align for something like this to be made in particular,

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as was just mentioned, the exhibition which remarkably,

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the exhibition that started in Scotland earlier this year was the first ever major retrospective of its type in this country.

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And it's amazing there's only one Escher work here other than that which is in the hunterian in Glasgow.

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Amazing that now thanks to the collection at the Community Museum, so many are here and I would strongly recommend the,

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the exhibition which just moved to Dulwich and is going to be there till January,

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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.

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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.

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And you can see in the works, outside works,

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it was so familiar with reproductions on prints and posters and on album covers actually to see the craft,

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the the woodcraft scale that's needed to make those images.

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I think that's something that really surprised us. I too have a couple of thank you's before I start.

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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.

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So I don't think he's here to put a hand that these were made, especially for us.

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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,

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were were initially used for the models that the professor and his father originally constructed,

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and they were supplied by Shirley Hodgson, physicist as a thank you for that.

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And finally, thank you very much,

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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.

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Thanks very much indeed. Enjoy the film. Thank you. Thank you.

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Thank you. So the picture I want to tell you about. Find where to stand.

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It's this one which made a very brief appearance in the movie, but wasn't really discussed at all.

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And it's English name is Print Gallery, and it's got some very interesting mathematics behind it,

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which was uncovered in about 21, 22 by a Dutch mathematician called Hendrik Leinster.

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And I left or came and gave us a talk in 2002 on it.

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And I want to share with you what I learned from him that day about the mathematics behind this print gallery.

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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.

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And if you follow it around to the top right, then the buildings get bigger.

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And eventually you realise that the building that he's stood in is a building that's in the

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town that he's looking at so that he is standing in the picture that he is looking at.

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So you have this cyclic expansion as you go around.

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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.

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And it carries on going around and around never ending.

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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.

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There's nothing unusual about it and it doesn't matter which bit you look at.

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It looks like a perfectly normal bit of a print.

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It's only when you put it all together that you get the illusion because this expansion is gradual as you go around.

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So the first thing I want to tell you is how how I should generated this picture

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and then I want to tell you what the mathematicians did when they got hold of it.

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So you started with the idea of laying a grid of squares down on a piece of paper,

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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.

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And the first thing you tried to do was to use straight lines to do this.

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So you had this fan of lines coming out of the corner, so that if you start with them a certain distance apart here,

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by the time you get to this edge, they're four times as far apart.

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And if you do that, then the little square that you shade down in the bottom right becomes four times as big.

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By the time it gets to this corner.

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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,

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you've increased your scale by a factor of four times, four times four times four, which is 256.

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But the problem with using straight lines is where you can see it in this picture.

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I started off with a little square and I've ended up with a shape in that corner,

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which is not a square anymore because the lines are not at right angles anymore.

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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.

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So that if you did look a little bit of this, it wouldn't look like a normal print.

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It would be too distorted. So I started this way and then threw it away and decided not to do this.

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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,

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which were at right angles to each other, would stay at right angles to each other everywhere.

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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.

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And he just found this this idea, I think intuitively in mathematics,

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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.

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And that was the conformal map of my title. So the actual map that you use looks something like this.

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So you can see that if you look at any little bit of this, it just looks like a grid of squares.

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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.

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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.

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So by the time you've gone all the way around this picture, one little square would become 256 times as big.

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So that's the grit that he got. How then did you do you make the picture once you've got the grit?

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Well. So you take your original unreformed picture, you do a sketch, and you lay this grid of squares over the top of it.

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And then what you have to do is that for every square up here, you look to see what's in that square,

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and you copy it out onto the same square, into the deformed picture.

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And when you do that and you do it assiduously for each one of these squares,

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where then that will automatically give you this expansion, the zooming in as you go around.

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So that was the way that Asha generated his picture and led him to this by the time you filled in all the squares.

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So Leinster was fascinated by this and you asked himself two questions.

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So the first question was what did the original picture look like that actually started with before it was deformed?

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This is what the deformed one looks like. Can you reproduce the original rectified picture?

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The one that would still have straight lines? And then the second question was, why did you leave a hole in the middle here?

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What should go in the hole? And if we know what the what the grid is, could we continue the picture into the hole?

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And what would happen if we did continue it into the hole? So I'm going to try and answer both those questions.

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So the first thing that Lance did to try and get some information about about what

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exactly it was that Asher had done is we had this transform grid over here on the left,

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and then the original picture is somewhere here on the right.

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I don't know what it looks like yet, but let me assume the picture is sat behind here somewhere.

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And I'm going to walk around in this transform picture and see where I end up with in the original picture.

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So I'm first going to do a path where I walk from A to B.

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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,

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and then I'm going to turn left and go to see.

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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.

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So one big square here becomes a square, which is very small here.

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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.

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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.

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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.

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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.

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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,

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because I got this factor of four for every time I walked along with one of these edges.

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So given that in the original picture, I'm back where I in the transform picture, I'm back where I started.

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That means in the original picture, whatever is at this position must be the same as it's at this position.

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I must have a copy of it so that I got back to where I started.

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And that tells you that whatever this original picture is, it's invariant by scaling by 256.

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So that if you took this picture and you blew the whole thing up by a factor of 256, it would look the same.

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So this is known more or less universally now is the Droste effect.

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After a Dutch brand of cocoa powder. So so this is a very famous cocoa powder in Holland.

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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.

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So she's carrying the tray. That is the box. That is her again.

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And of course, on that box, there's a picture of her carrying the box.

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And on that box, that's a picture of her carrying the box and so on, ad infinitum.

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So this picture carries on. It gets very small because of the factor of about five each time.

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But it's invariant in the sense that if you blew this box up so that it was the size of this box,

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then that little box would be this box and etc., etc., and you would picture would look the same again.

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And I can give you a live demonstration of this if this camera is working.

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This is the sort of thing that it's called video feedback. So is you.

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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.

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And it carries on all the way down. So whatever the original picture was, it has this invariance by a factor of 256.

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Okay, let's do another walk. So I'm going to start again from a at this time I'm going to walk.

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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,

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and now I'm going to turn left and I'm going to work eight squares again.

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The squares are getting smaller, so I don't get quite so far.

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I'm going to work eight squares down and I'm going to turn left again and walk eight squares to the right.

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So now in the original picture, I've got back to where I started, but in the deformed picture I haven't.

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I've got to some point here and the same argument works again now.

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So because the original picture I must have the same thing a day,

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then must be the same thing at this point as there is at this point in that transform picture.

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And that immediately answers the question as to what should go in the middle of the whole.

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That this is this point here is effectively in the middle of the whole.

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So what they should be there is exactly the same thing that you see.

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But shrink and I have to rotate it a little bit because there's been a bit of rotation here as well.

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So the transform picture as well is invariant under scaling and rotation.

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So in that little hole in the middle, there should be another copy of the whole thing shrunk down and rotated.

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And of course, in that copy,

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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.

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So I'll show you a demonstration later on where I have a version of this picture

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where we don't take the whole out to show you what really goes in the middle. Okay.

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So I'm going to demonstrate the map in a different way now.

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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.

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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.

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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.

189
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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.

190
00:19:44,000 --> 00:19:46,370
So if I blew this picture up by a factor of four,

191
00:19:46,400 --> 00:19:50,450
it would look the same because this grid would become that one and the middle one would become this one.

192
00:19:50,720 --> 00:19:53,570
And then the one I hadn't drawn in the middle would come and fill out the inner one.

193
00:19:55,460 --> 00:19:59,690
And just to show you so you can keep track of which grid is which I'm going to colour code them.

194
00:20:00,110 --> 00:20:03,019
So what you have to imagine is that if I draw a picture on this, whatever,

195
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I draw on the green, I draw the same thing on the blue, but scaled by a factor of four.

196
00:20:07,880 --> 00:20:11,090
And I draw the same thing again on the red but scale by a factor of four.

197
00:20:11,750 --> 00:20:16,580
So this picture carries on all the way down. And then I want to show you what Asha's map does.

198
00:20:16,730 --> 00:20:24,650
It cuts this picture along that the dark line that you can see there, and it shifts one side relative to the other.

199
00:20:26,640 --> 00:20:32,850
One is getting slightly bigger, ones getting slightly smaller, and then you stop when it exactly matches up again.

200
00:20:33,660 --> 00:20:41,580
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.

201
00:20:41,880 --> 00:20:43,830
And the blue is a perfect copy of the green.

202
00:20:44,280 --> 00:20:50,730
And the size has been shifted so that it exactly matches so that instead of this thing now being invariant as you go in,

203
00:20:50,760 --> 00:20:53,760
I have this you can sort of see the spiral expansion now.

204
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But as you go around, things get smaller, but it joins up perfectly.

205
00:20:58,870 --> 00:21:05,670
And you can also see on this picture that the red is a copy of the blue, which is a copy of the green.

206
00:21:05,680 --> 00:21:07,960
So you can see, because I've only got a factor of four,

207
00:21:08,350 --> 00:21:14,630
you can see that this picture repeats itself still as I go in, but the green doesn't join up to green anymore.

208
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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.

209
00:21:25,240 --> 00:21:30,850
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.

210
00:21:31,810 --> 00:21:36,180
So how would you actually work out mathematically how I generated this picture?

211
00:21:36,190 --> 00:21:41,920
How did I know what map to use? So to do that, Leinster divided the map into three stages.

212
00:21:43,000 --> 00:21:46,210
So the first stage is to take a logarithm.

213
00:21:46,360 --> 00:21:52,060
So let me show you what a logarithm does to this picture. It opens it up along that cut to gain.

214
00:21:53,280 --> 00:21:58,590
But this time it opens it out and folds it back. And you get something like this.

215
00:22:00,720 --> 00:22:05,600
So the. Let me explain what the act is.

216
00:22:05,720 --> 00:22:11,000
So if in the original picture I can describe any point in this picture by using two numbers,

217
00:22:11,000 --> 00:22:15,020
one is the distance from the centre and one is a bearing or the angle.

218
00:22:16,220 --> 00:22:21,320
And when you take the logarithm of that picture, what you find is that on this axis you get the angle.

219
00:22:22,370 --> 00:22:29,089
And on this axis, you get the logarithm of the distance. So the three things to notice now about this picture.

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00:22:29,090 --> 00:22:36,350
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.

221
00:22:36,830 --> 00:22:38,780
So little squares are still little squares.

222
00:22:40,550 --> 00:22:48,260
The second is that when I do this angle, I've only shown you the angle going between, well, -182 plus 180 here.

223
00:22:48,680 --> 00:22:51,710
But of course, you can carry on around as many times as you want in this.

224
00:22:52,310 --> 00:22:59,750
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.

225
00:23:00,260 --> 00:23:05,329
So really, this picture goes on, carries on, going upwards there, but it's periodic.

226
00:23:05,330 --> 00:23:13,880
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.

227
00:23:14,390 --> 00:23:17,960
So it has this. You can slide it and it looks the same.

228
00:23:19,490 --> 00:23:24,350
And that the reason for taking a log is that this now does the same thing in the other direction as well.

229
00:23:24,710 --> 00:23:29,570
So with this original picture, if you multiply by four, you get back to where you started.

230
00:23:30,380 --> 00:23:31,550
But once you take a log,

231
00:23:31,760 --> 00:23:39,500
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.

232
00:23:40,040 --> 00:23:45,560
So by taking a log, you turn a scale invariance into a translational invariance.

233
00:23:45,950 --> 00:23:52,189
So in this picture, multiplying by four corresponds to adding log for to the log R.

234
00:23:52,190 --> 00:24:01,610
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,

235
00:24:01,610 --> 00:24:05,540
but it's also periodic in this direction. And that's the first step.

236
00:24:07,590 --> 00:24:12,360
Okay. That's the hard step. The second step is to rotate this picture.

237
00:24:13,940 --> 00:24:16,069
So I've added a little bit more on the top and the bottom.

238
00:24:16,070 --> 00:24:22,130
So you can see it would carry on periodically and now I'm going to rotate it and scale it a little bit.

239
00:24:23,370 --> 00:24:30,870
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.

240
00:24:31,980 --> 00:24:39,720
And by doing that, this picture is still periodic in the direction with the same period of 360 degrees.

241
00:24:41,130 --> 00:24:45,270
Because the blue is a copy of the red and the green is a copy of the blue.

242
00:24:45,660 --> 00:24:51,510
So I've got the same invariants. It's still got the same translation invariants in the vertical direction.

243
00:24:52,290 --> 00:24:59,250
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.

244
00:25:00,420 --> 00:25:04,320
So when you undo it, so it stitches it back together again.

245
00:25:05,220 --> 00:25:10,420
But now instead of blue joining on onto blue and green. Joining on to green, because I did that rotation.

246
00:25:10,920 --> 00:25:15,090
Now green joins onto blue and blue joins on to red.

247
00:25:15,330 --> 00:25:22,260
And that's how you make this map. So the key thing was, was getting that rotation right in the logarithm plane.

248
00:25:22,260 --> 00:25:27,510
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.

249
00:25:30,960 --> 00:25:32,280
So let me show you.

250
00:25:32,820 --> 00:25:40,320
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.

251
00:25:41,460 --> 00:25:44,760
And this is what it looks like. So this is the map without any distortion.

252
00:25:46,500 --> 00:25:49,770
And you can see that everything looks perfectly normal. But you might ask.

253
00:25:49,770 --> 00:25:55,649
Well, well. So first of all, what's this bit missing? That's the bit that fell outside the boundaries of this painting.

254
00:25:55,650 --> 00:26:01,860
So we don't know what should be that didn't draw it. But you might also ask, well, where's the town gone?

255
00:26:01,890 --> 00:26:07,470
Where's the ship gone? Where's the rest of it? And of course, I've only shown you this at one scale.

256
00:26:08,070 --> 00:26:11,790
This is invariant under scaling by 256.

257
00:26:11,790 --> 00:26:15,030
So I've got to zoom in a bit to the middle to show you what the rest of it is.

258
00:26:15,810 --> 00:26:23,400
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.

259
00:26:23,820 --> 00:26:28,160
You start to see the rest of the town and you zoom in by a factor of four again.

260
00:26:28,200 --> 00:26:34,110
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.

261
00:26:34,680 --> 00:26:41,100
There's another guy and you can keep going round. It's invariant under zooming in by 256.

262
00:26:44,560 --> 00:26:53,830
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,

263
00:26:53,830 --> 00:27:00,250
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?

264
00:27:01,390 --> 00:27:06,550
And it took a little while before technology caught up with my aspirations.

265
00:27:07,870 --> 00:27:10,989
But when we finally moved into the new building, I had the opportunity.

266
00:27:10,990 --> 00:27:19,150
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.

267
00:27:20,110 --> 00:27:28,060
So we went off and took some pictures. So I updated it a little bit instead of having a guy in a print gallery.

268
00:27:28,540 --> 00:27:31,150
I had a guy at a desk looking at a computer screen.

269
00:27:34,270 --> 00:27:40,280
And then Alan and I went to Green College Town next door, and we took this nice picture of the Mathematical Institute.

270
00:27:41,210 --> 00:27:46,420
And the reason it's hard to do this is that one picture is not good enough,

271
00:27:46,460 --> 00:27:51,680
because if you do one picture and you zoom in by 256, you just lose all your resolution.

272
00:27:51,680 --> 00:27:54,920
Even if it's a 16 megapixel picture, it just doesn't work.

273
00:27:55,430 --> 00:27:58,940
So you need to take multiple pictures and they need to really fit together nicely.

274
00:27:59,540 --> 00:28:02,389
So you zoom in on this one. My office is just there.

275
00:28:02,390 --> 00:28:09,770
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.

276
00:28:10,220 --> 00:28:15,049
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,

277
00:28:15,050 --> 00:28:18,590
you just see windows look like that. You don't see people like that in the middle.

278
00:28:18,600 --> 00:28:23,749
So you have to do a little bit of Photoshop and then you zoom in on this and you see the

279
00:28:23,750 --> 00:28:26,750
guy at the computer screen and then you zoom in on this again and you get back to that.

280
00:28:26,760 --> 00:28:31,790
So you have it. It's cyclical again. And in fact, now that I've got pictures and a computer,

281
00:28:31,790 --> 00:28:38,690
I can really show you what this dress effect looks like by turning this into a little movie where you zoom in.

282
00:28:41,470 --> 00:28:44,740
And it keeps going, as in this guy.

283
00:28:48,520 --> 00:28:56,200
And of course, it'll keep going for as long as I show it. Okay.

284
00:28:56,200 --> 00:28:59,920
So I've got my four pictures so I could a whole lot of each one.

285
00:29:00,490 --> 00:29:05,860
And then the idea is that so this picture fits in that whole half the scaling and this picture

286
00:29:05,860 --> 00:29:10,030
fits exactly in that whole after scaling and this one and that whole and then of course,

287
00:29:10,030 --> 00:29:19,510
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.

288
00:29:20,110 --> 00:29:26,709
So even though each of those pictures was a factor of four smaller than the others,

289
00:29:26,710 --> 00:29:30,040
so the relative to the original picture, there are very small part of it.

290
00:29:30,610 --> 00:29:34,120
Each one contributes an equal amount to the final image.

291
00:29:35,020 --> 00:29:41,620
And so if I've got those exactly right, these should all line up when you put the four bits together.

292
00:29:43,890 --> 00:29:48,720
It clicks and you get the actor image, which is outside.

293
00:29:57,590 --> 00:30:01,430
And our building really lends itself to this picture because the nice straight lines you see,

294
00:30:01,430 --> 00:30:10,130
the flowing curves that assured also wanted to do in his picture by making the print gallery with a row of prints.

295
00:30:10,670 --> 00:30:18,560
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.

296
00:30:19,430 --> 00:30:22,069
And we blew this up and there's a picture of this outside the door.

297
00:30:22,070 --> 00:30:24,950
You might have seen it on your way in, but you can certainly have a look on your way out.

298
00:30:27,710 --> 00:30:31,100
Now, so I said for my picture, I haven't taken the middle out.

299
00:30:31,520 --> 00:30:34,879
So in the middle there, the scale of 256,

300
00:30:34,880 --> 00:30:42,380
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.

301
00:30:43,070 --> 00:30:51,920
And as you zoom in, you get the same thing back again. This.

302
00:31:01,470 --> 00:31:08,370
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

303
00:31:08,370 --> 00:31:13,110
is and some of the things that Escher might have played with if it's still been around.

304
00:31:13,770 --> 00:31:18,810
So let me just go back to how we generated this map. We we had this periodic structure.

305
00:31:20,820 --> 00:31:29,040
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.

306
00:31:29,370 --> 00:31:34,649
So this is one period. So these are like the tiling. So each one of these squares is repeated.

307
00:31:34,650 --> 00:31:37,680
So whatever's in this square gets repeated in every other square.

308
00:31:39,240 --> 00:31:47,070
And then the way I generated the map was to rotate this and scale it so that this point light exactly above this point.

309
00:31:49,470 --> 00:31:55,410
So I chose this red point and I turned it into a point that lies exactly above the one that was fixed.

310
00:31:56,130 --> 00:31:59,700
But the map would work if I chose any other point here.

311
00:31:59,910 --> 00:32:04,170
There's no special reason to choose this one. So I could choose any point here.

312
00:32:04,470 --> 00:32:09,780
Rotate in scale and put it at the position of this red one and the picture would join up.

313
00:32:10,140 --> 00:32:16,740
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.

314
00:32:17,680 --> 00:32:24,010
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.

315
00:32:24,310 --> 00:32:25,530
This one is also interesting.

316
00:32:25,540 --> 00:32:33,340
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.

317
00:32:33,730 --> 00:32:38,920
It's just that now it expands as you go anticlockwise and gets smaller as you go clockwise.

318
00:32:39,580 --> 00:32:43,240
But it's also something that that you could equally have drawn.

319
00:32:43,960 --> 00:32:47,320
It's rather unfortunate because of the scale factor as you go up the side.

320
00:32:47,330 --> 00:32:52,570
So this guy has got a very big head. This guy's got a very little head on which one is better.

321
00:32:53,920 --> 00:32:58,330
And various other ones have these. You see that they get they get rather strange.

322
00:32:58,570 --> 00:33:01,660
This one has got it expands twice as you go around.

323
00:33:01,660 --> 00:33:03,280
So you have an even stronger factor.

324
00:33:03,760 --> 00:33:09,970
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.

325
00:33:10,000 --> 00:33:13,420
But maybe that's not such. It's something that we're in actually too much.

326
00:33:13,420 --> 00:33:17,650
And some of his other prints. Okay.

327
00:33:17,780 --> 00:33:24,079
And the last thing I wanted to say was this number 256 that's come up all the way through.

328
00:33:24,080 --> 00:33:31,280
Right. That there was a factor of 256 invariants in the picture you scale by 256, you get the same thing again.

329
00:33:31,310 --> 00:33:40,100
So was there anything special about 256 Y 256 and now that I have these images,

330
00:33:40,100 --> 00:33:44,540
I can change what the scale factor is just by changing the size of the image I'm looking at.

331
00:33:45,140 --> 00:33:52,160
So if I zoom in at the picture on the screen, I'm changing the scale factor in the drastic effect.

332
00:33:53,240 --> 00:33:58,370
And you see 256, you don't really notice really it's a Droste effect till you get close to the end where

333
00:33:58,370 --> 00:34:03,050
you get to this is a factor of four and now suddenly you can see multiple heads here.

334
00:34:03,350 --> 00:34:07,820
So you see that you are looking at a picture that has this video feedback, this Droste effect.

335
00:34:09,650 --> 00:34:14,570
So from here I can show you what the map looks like if I want to ask this.

336
00:34:18,050 --> 00:34:21,430
Didn't do very smoothly. We'll see if the computer can do that again.

337
00:34:24,670 --> 00:34:28,080
Yeah. So you shift the head along. Right.

338
00:34:28,090 --> 00:34:32,110
You choose one head and you're shifted along and it joins on to the next body.

339
00:34:33,250 --> 00:34:39,700
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.

340
00:34:39,700 --> 00:34:42,340
And then inside again, there's a smaller version all the way down.

341
00:34:43,300 --> 00:34:46,930
So this picture also looks interesting, but it looks very different to the original.

342
00:34:47,440 --> 00:34:51,370
So a factor of four is not very good for generating that original illusion.

343
00:34:51,370 --> 00:34:57,610
I think sort of central to this illusion was that you didn't know that it carried on infinitely, often into the middle.

344
00:34:57,880 --> 00:35:02,500
You didn't know what was there. You only saw one version. It's here. You clearly see multiple versions.

345
00:35:04,810 --> 00:35:09,820
And now I can scale back out again so I can show you what the actual picture would look like at various different scales.

346
00:35:10,840 --> 00:35:15,360
So. Let me see if I can pause it a few times.

347
00:35:15,540 --> 00:35:20,320
You scale out. Gradually the guy in the middle gets smaller.

348
00:35:20,920 --> 00:35:27,070
You start to see more of the picture. You've got to scale out a long way before you're looking at something interesting.

349
00:35:27,580 --> 00:35:31,810
But all of them work. They would have a bigger white dot in the middle.

350
00:35:32,780 --> 00:35:35,300
And then you finally get to the original one.

351
00:35:36,140 --> 00:35:44,150
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.

352
00:35:44,250 --> 00:35:47,840
It was too small. Then you would really I think you'd have a big gap in the middle,

353
00:35:48,200 --> 00:35:51,890
or you would see multiple copies of the image and it would be a different type of picture.

354
00:35:53,510 --> 00:35:59,180
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.

355
00:36:00,740 --> 00:36:05,090
There's a paper that Leinster and De Smet wrote on the work that they did that's available.

356
00:36:05,270 --> 00:36:11,450
They have a nice website where they have lots of images of the maps that they that I've shown you.

357
00:36:11,930 --> 00:36:18,830
And then some of the pictures that showed how you generated this in the first place are from the book by Bruno Ernst.

358
00:36:19,810 --> 00:36:30,680
Okay. Well,

359
00:36:30,680 --> 00:36:39,020
it's a great pleasure to be able to express my appreciation of Escher and to try and tell

360
00:36:39,020 --> 00:36:43,460
you a few things which perhaps he would have done things with if he'd lived longer.

361
00:36:44,360 --> 00:36:51,920
Well, that, of course, is another one. But I want to start by talking about the tiling outside the main building here.

362
00:36:52,610 --> 00:36:56,240
You saw that in the film and you saw it as you came in the door.

363
00:36:58,220 --> 00:37:02,990
And that is the kind of thing that Escher might have made use of.

364
00:37:03,050 --> 00:37:09,140
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.

365
00:37:11,240 --> 00:37:15,820
Let's see. This works. You may well have seen this picture.

366
00:37:15,860 --> 00:37:19,340
It's just one example of an Escher, but it's one of the ones outside the door.

367
00:37:20,360 --> 00:37:23,240
And it's an example illustrates the symmetry.

368
00:37:23,450 --> 00:37:30,910
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.

369
00:37:30,920 --> 00:37:35,240
It's what's called a glide symmetry. That's one example.

370
00:37:37,580 --> 00:37:41,180
Now I'll show you another example which you saw in the film.

371
00:37:41,630 --> 00:37:46,280
Now, this one has the property that it has rotational symmetry.

372
00:37:46,610 --> 00:37:51,740
Now you see that it has points. Well, here's a point of twofold symmetry.

373
00:37:51,920 --> 00:37:57,770
So you rotate that one so you can rotate through 180 degrees and the whole picture goes into itself.

374
00:37:59,210 --> 00:38:04,610
And here's a point of sixfold symmetry, and then you have points of three for symmetry.

375
00:38:05,270 --> 00:38:09,469
So if I can find one in the middle here somewhere. There.

376
00:38:09,470 --> 00:38:16,460
I think there's one. So he exhibited some of the crystal symmetries that, if possible, here.

377
00:38:16,790 --> 00:38:23,660
Now, when I say crystal symmetry, I mean something which can be associated with a translation of symmetries.

378
00:38:24,170 --> 00:38:27,590
And those symmetries are exactly as illustrated here.

379
00:38:28,040 --> 00:38:33,680
We have the two fold symmetry that we just saw and the three fold and the six fold, but also four fold.

380
00:38:34,610 --> 00:38:37,790
So I should say that that's four fold and they're six fold.

381
00:38:38,390 --> 00:38:50,299
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.

382
00:38:50,300 --> 00:38:55,140
That's an endowment shape. And this half star just as cap over here.

383
00:38:55,740 --> 00:39:00,290
It just uses this for shapes. And this thing created the pattern.

384
00:39:00,300 --> 00:39:05,720
Well, I think I'll circumvent this legal problem by showing a different version of it.

385
00:39:06,920 --> 00:39:14,090
So you can see the pattern, the larger area of the pattern, and it's made up of just of those shapes.

386
00:39:14,370 --> 00:39:19,610
If you just look at it casually, it has a look as though it is has a translational symmetry.

387
00:39:19,620 --> 00:39:24,710
It looks almost as though it's a crystal pattern. However, and here's why I would need to show you the whole thing,

388
00:39:25,010 --> 00:39:30,860
because it looks like a Pentagon on the outside, and in fact, it has a five fold symmetry.

389
00:39:31,250 --> 00:39:36,860
And that's not one of the crystal symmetries. Now, that's a theorem that you can only have those ones I told you.

390
00:39:37,220 --> 00:39:46,490
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.

391
00:39:47,090 --> 00:39:52,460
Now, I want to show you how this pattern is constructed. So let's go back to my previous one.

392
00:39:52,910 --> 00:40:06,170
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.

393
00:40:06,320 --> 00:40:15,440
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.

394
00:40:15,470 --> 00:40:24,710
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.

395
00:40:25,100 --> 00:40:31,610
And so there we have a bigger pattern. I think I'll leave this one here because I want to take another stage.

396
00:40:32,780 --> 00:40:41,900
There we have the next stage. Those orange ones are again collected together to form these big new ones.

397
00:40:42,830 --> 00:40:50,580
And the blue ones are not here. I have a problem because. I won't get that all that good.

398
00:40:51,030 --> 00:40:58,080
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.

399
00:40:59,010 --> 00:41:03,540
Now, the hierarchy is not a thing that you sort of notice.

400
00:41:03,750 --> 00:41:07,350
Look at the pattern. It looks very irregular and you don't really see any hierarchy.

401
00:41:07,350 --> 00:41:10,020
You see a regularity about it, which is very strong.

402
00:41:11,610 --> 00:41:16,349
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,

403
00:41:16,350 --> 00:41:20,850
but here, take one of these lines and you follow that line all the way along and it keeps on going.

404
00:41:21,540 --> 00:41:26,010
Any line in the picture, whichever direction it's pointing, it, it just keeps on going.

405
00:41:26,490 --> 00:41:35,550
So I guess it's hard for me to watch that and this at the same time, but moreover, it has interesting sub patterns.

406
00:41:36,060 --> 00:41:45,180
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.

407
00:41:46,020 --> 00:41:52,739
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.

408
00:41:52,740 --> 00:42:00,420
And they're always like that everywhere. Whenever you find one of those is always surrounded by a ring of ten pentagons.

409
00:42:00,540 --> 00:42:06,720
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.

410
00:42:07,890 --> 00:42:12,900
So there's a lot of order in this thing, which is nothing to do really with the hierarchical arrangement.

411
00:42:13,260 --> 00:42:18,149
It just has this kind of local order,

412
00:42:18,150 --> 00:42:24,750
which is which is highly symmetrical and which in a sense shouldn't be because it's one of forbidden crystal symmetries.

413
00:42:25,470 --> 00:42:29,880
Now, I don't know where I can show you this, but I can I can show you them one at a time.

414
00:42:30,480 --> 00:42:36,200
Here I have you see, this is just a mathematical way in which you can construct this picture.

415
00:42:36,210 --> 00:42:40,830
But I had the idea is maybe you can force that arrangement with a kind of jigsaw puzzle.

416
00:42:41,670 --> 00:42:46,409
So here we have six pieces. Well, there are three pentagons.

417
00:42:46,410 --> 00:42:56,820
That one. That one. And that one. And we have these pentacle, the Justice Cup and the and the rhombus.

418
00:42:58,020 --> 00:43:02,310
They are decorated with these little knobs and things. So that's how you make it into a jigsaw puzzle.

419
00:43:02,670 --> 00:43:06,690
You have to fit the little knobs in the right holes and so on.

420
00:43:08,580 --> 00:43:15,180
Six. What the [INAUDIBLE] are you doing?

421
00:43:17,450 --> 00:43:20,870
Excellent. That does help. Thank you.

422
00:43:21,500 --> 00:43:30,800
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,

423
00:43:31,430 --> 00:43:38,390
then they will fit together only in the kind of way I've shown you, which will be almost periodic, but never quite.

424
00:43:39,020 --> 00:43:43,849
So it's almost a crystal, but not quite. It's the sort of thing people call quasicrystals.

425
00:43:43,850 --> 00:43:49,280
And actually they now find actual substances which which have this kind of symmetry and

426
00:43:49,280 --> 00:43:53,210
the atomic arrangement seem to follow the kind of thing that I have been showing you.

427
00:43:53,810 --> 00:43:59,900
So I had this is a sort of curious jigsaw or mathematical puzzle where you've got the six shapes,

428
00:44:00,230 --> 00:44:04,370
which will only fit together in a way which never repeats itself.

429
00:44:05,000 --> 00:44:15,500
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,

430
00:44:15,860 --> 00:44:24,560
which had been produced in 1971, based on some earlier ideas from how long and one of his students,

431
00:44:24,980 --> 00:44:30,740
and that he had reduced these original considerations to the six set of tiles.

432
00:44:31,130 --> 00:44:36,740
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.

433
00:44:36,740 --> 00:44:42,980
But they have the same property that they will tile the whole plane, but only in a way which never repeats itself.

434
00:44:43,640 --> 00:44:46,730
That's what's called a periodic set of tiles.

435
00:44:47,450 --> 00:44:54,710
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,

436
00:44:56,090 --> 00:45:03,229
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.

437
00:45:03,230 --> 00:45:05,930
And this one's got one of the spiky things and the only place it can fit in here,

438
00:45:06,050 --> 00:45:10,100
it's all I need to do is cut out this piece, stick it on there and stick it on there.

439
00:45:10,490 --> 00:45:16,550
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.

440
00:45:17,060 --> 00:45:21,590
And I eventually came up with two.

441
00:45:22,550 --> 00:45:27,800
Now, I remember feeling disappointed that you might think the strange reaction be disappointed because I

442
00:45:27,800 --> 00:45:35,270
thought it is so simple that surely it's well known and I just want to relate it to the original set.

443
00:45:35,570 --> 00:45:45,500
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.

444
00:45:45,500 --> 00:45:54,140
And dots, cuts of dots of every dot has the same pattern inside it and so does every cut.

445
00:45:55,040 --> 00:46:01,159
And if you fit the match them together in this way, then you can match the cuts.

446
00:46:01,160 --> 00:46:11,960
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.

447
00:46:12,530 --> 00:46:21,170
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.

448
00:46:22,520 --> 00:46:26,630
So that's done with two. So we've been talking about issue here.

449
00:46:27,080 --> 00:46:32,540
So I did wonder at one stations that should really be influencing me with his intriguing

450
00:46:32,540 --> 00:46:38,570
all sorts of ways of making periodic patterns into into interesting animals and so on.

451
00:46:38,870 --> 00:46:44,269
So I wondered, can you assure this? So I came up with one.

452
00:46:44,270 --> 00:46:48,500
This is not my first attempt. I'll show you that in a minute, but it's almost my first attempt.

453
00:46:49,040 --> 00:46:52,190
And you see, it's made up out of two bird shapes.

454
00:46:52,760 --> 00:46:56,899
There is the movie The Fat Bird, if I can find one.

455
00:46:56,900 --> 00:47:00,800
There's the fat bird there. And the thin bird, which is this one here.

456
00:47:01,460 --> 00:47:06,320
And the only way that they will fit together is in this never repeating pattern.

457
00:47:07,010 --> 00:47:14,480
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.

458
00:47:14,780 --> 00:47:19,610
I hope I have lined this up right. We have to get it like that.

459
00:47:19,850 --> 00:47:27,230
And I think you will see that the little birds, the darts and the big birds are the cuts.

460
00:47:27,280 --> 00:47:38,060
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.

461
00:47:38,510 --> 00:47:41,990
Now, it's actually been around this time, unfortunately, just a few years.

462
00:47:42,500 --> 00:47:48,710
But sometimes my father did and neither of them saw these things, which was a great disappointment to me.

463
00:47:49,340 --> 00:47:52,820
But nevertheless, I have some interest.

464
00:47:55,340 --> 00:48:00,620
Now, there's another way of doing the two tails, and in some ways even simpler.

465
00:48:01,250 --> 00:48:05,940
That's where these rumba shapes again, you must have matching rules. Of course, there are lots of ways of fitting.

466
00:48:05,940 --> 00:48:08,960
We're on this list together. You could just take one of them and just do it periodically.

467
00:48:09,200 --> 00:48:11,810
So you've got to have a rule which stops you doing that.

468
00:48:12,350 --> 00:48:18,290
Stuff, you've got to match it so that the spots is exactly the same spots that I had previously.

469
00:48:18,830 --> 00:48:25,160
And in fact it is exactly the same as what I showed you previously, but just rearranged a little bit.

470
00:48:26,240 --> 00:48:31,280
So that is the same tiling, but things cut apart a bit and reassembled.

471
00:48:32,150 --> 00:48:35,360
And so there we have the rhombus version with the matching rules,

472
00:48:35,540 --> 00:48:45,170
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,

473
00:48:45,830 --> 00:48:48,940
which is called a periodic set. Okay.

474
00:48:48,950 --> 00:48:54,740
Well, when I heard about this wonderful building as it was being constructed, Nick Woodhouse was one of the.

475
00:48:54,890 --> 00:49:04,190
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.

476
00:49:04,190 --> 00:49:04,910
So I thought, well, look.

477
00:49:05,780 --> 00:49:13,790
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.

478
00:49:14,450 --> 00:49:24,080
And if you remember going back to the version I have here, there were these rather nice rings, these decorations.

479
00:49:24,410 --> 00:49:28,820
And every time you have a deck, again, it's surrounded by a ring of ten pentagons.

480
00:49:29,390 --> 00:49:34,070
So let's think about those rings and I'll mark the rings.

481
00:49:34,910 --> 00:49:38,200
Actually, it's better if I go to the next stage of the hierarchy.

482
00:49:38,230 --> 00:49:48,490
So. So let's go to the orange ones there. And I mark those rings of just around the deck decorations.

483
00:49:49,070 --> 00:49:56,780
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.

484
00:49:56,780 --> 00:49:59,870
And that happens all the way over. So you can put those rings down.

485
00:50:00,440 --> 00:50:06,050
Oh, that would make a nice pattern because it brings out this rather attractive feature of those rings.

486
00:50:06,860 --> 00:50:11,720
Of course, just doing that, we have a lot of gaps all over the place.

487
00:50:12,140 --> 00:50:21,880
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.

488
00:50:21,950 --> 00:50:24,440
Yes, there were a few more lines on this.

489
00:50:24,890 --> 00:50:33,380
So you see, now that I have a sort of curvilinear version of what I had before, the the the here's the pentacle.

490
00:50:33,410 --> 00:50:37,130
You see, if you follow the green lines around, it's a sort of curved version of it.

491
00:50:38,000 --> 00:50:43,370
And the rhombus as a curved version of the Rhombus and the Justice Cup, there's a curved version of it.

492
00:50:43,850 --> 00:50:46,700
And then the Pentagon's get distorted in various different ways.

493
00:50:46,700 --> 00:50:52,309
But the three different versions of the Pentagon is distorted in different ways anyway.

494
00:50:52,310 --> 00:51:01,450
So there you are. It makes a nice pattern, but what does that do when you go to the to the rhombus?

495
00:51:01,550 --> 00:51:07,220
It just disappeared on top of it.

496
00:51:08,300 --> 00:51:13,850
Go to the rhombus is instead to take away the Pentagon's because that confuses

497
00:51:13,850 --> 00:51:21,920
restricting but on business there and now we have a way of marking the embassies.

498
00:51:22,730 --> 00:51:28,130
However, we still have a lot of gaps. Some of them are different, so we better fill those gaps in, too.

499
00:51:29,150 --> 00:51:33,740
And when I've done that, I can find the right way around this thing.

500
00:51:34,700 --> 00:51:41,510
Yeah. Now every fat rhombus has the same crossed green lines on it.

501
00:51:42,170 --> 00:51:49,130
The everything about rhombus has two arcs on it, and they fit together to make this pattern.

502
00:51:49,250 --> 00:51:54,559
And this is exactly the pattern you find outside the building, except that it isn't green,

503
00:51:54,560 --> 00:52:03,860
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.

504
00:52:04,280 --> 00:52:07,850
So if you want to know what it is, that's what it is. Yes.

505
00:52:08,450 --> 00:52:15,650
Okay. Now, let me say something more about Sharon's issues.

506
00:52:16,250 --> 00:52:19,490
In fact, I think I'm going to go to the other device now.

507
00:52:20,140 --> 00:52:24,200
I could thank you, except it hasn't done it.

508
00:52:25,100 --> 00:52:29,060
Oh, yeah, no, that's not the first one. Oh, can I go to the first one.

509
00:52:30,860 --> 00:52:37,250
Yeah. This is actually the first version Escher ization I did which appeared anywhere in the article I had.

510
00:52:37,910 --> 00:52:43,970
And you can see that it's the cuts in the dots really, but slightly distorted to make two birds.

511
00:52:45,200 --> 00:52:50,779
Now I want to show you another one.

512
00:52:50,780 --> 00:52:52,160
So let me move this on.

513
00:52:52,700 --> 00:53:03,260
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.

514
00:53:03,590 --> 00:53:11,360
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.

515
00:53:11,360 --> 00:53:17,560
Might well have done. The bullfrogs are actually catching ducks.

516
00:53:18,130 --> 00:53:22,420
And I think I'm going to have to come back to this screen.

517
00:53:23,350 --> 00:53:26,990
If you do it for me, it'll save me from ruin everything. So here we are.

518
00:53:27,100 --> 00:53:31,540
Good. Thank you. This is the version of the Richard Hassel picture.

519
00:53:32,230 --> 00:53:35,320
And just to show you that, it usually just cuts dots.

520
00:53:36,640 --> 00:53:44,620
Here we go. And I think if I've got it in the right place, you'll see that the big frogs,

521
00:53:45,490 --> 00:53:50,110
they're very are really got a very distorted you see there, their arms and legs go way out in again.

522
00:53:50,770 --> 00:53:56,050
But where they all come together with the arms and the leg come together are the vertices of this picture.

523
00:53:56,380 --> 00:53:59,530
And this big white one here is a kite.

524
00:54:00,160 --> 00:54:03,270
This one is a cut. That brown one is cut.

525
00:54:04,120 --> 00:54:07,180
It's dark brown. One is a dart here.

526
00:54:07,570 --> 00:54:09,550
And you see this white one is a dart and so on.

527
00:54:09,820 --> 00:54:20,410
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.

528
00:54:21,700 --> 00:54:25,420
So I'm sure wish you would have done more things other people have done others.

529
00:54:26,170 --> 00:54:33,210
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.

530
00:54:33,730 --> 00:54:38,469
In fact, that was just the tilings that, you know, say something about.

531
00:54:38,470 --> 00:54:43,150
Oh, before I go onto sorry, I going to show it's not just another picture.

532
00:54:43,870 --> 00:54:50,590
This is a I just want to show you that the five fold one is just one of several.

533
00:54:51,130 --> 00:54:57,130
This is a 12 fold one. It's really very attractive due to go and then listen.

534
00:54:58,180 --> 00:55:05,270
There's four fold ones. So I should say eightfold ones, 12 fold ones and other well, seven fold ones look rather horrible.

535
00:55:05,290 --> 00:55:09,550
But the but the the eights and the 12, I think are very beautiful, particularly the twelves.

536
00:55:09,940 --> 00:55:17,050
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.

537
00:55:18,940 --> 00:55:31,150
Okay. Now I want to move on to something else that's very important doing, which is the impossible objects.

538
00:55:32,560 --> 00:55:42,070
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.

539
00:55:42,250 --> 00:55:47,910
Is it impossible or not? No. I guess we won't be doing the camera doing.

540
00:55:47,950 --> 00:55:53,230
I hope this is going to work, because it really depends on nobody having juggled this table.

541
00:55:55,150 --> 00:56:03,880
Nobody having job of this table. There was a lecture in here and then we had other lectures and wow, look, it still works.

542
00:56:05,110 --> 00:56:09,610
That was lined up just right. So you have a possible impossible triangle.

543
00:56:10,600 --> 00:56:15,850
And of course, if you want to see what it really is, you give us a little twist.

544
00:56:16,210 --> 00:56:19,480
Well, first of all, I think the best expert for this we had.

545
00:56:19,480 --> 00:56:24,250
There we are. Oh, we give it a little twist and then you can see what it is.

546
00:56:26,190 --> 00:56:31,470
And I think that's a very nice model made by my cousin Tony.

547
00:56:32,430 --> 00:56:35,940
And my father made one of these a long time ago.

548
00:56:38,040 --> 00:56:38,939
I don't know where it's got to.

549
00:56:38,940 --> 00:56:47,820
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.

550
00:56:48,180 --> 00:56:56,220
So. So, Tony, this really not very nicely done on this with very little indication of what it actually looks like, I think.

551
00:56:57,060 --> 00:57:00,780
Okay. So that's that one. No, I think I can take this away.

552
00:57:01,470 --> 00:57:05,530
And it's the stepchild next, isn't it? Yes. No.

553
00:57:05,590 --> 00:57:09,890
Why don't I take that off? Yes.

554
00:57:09,990 --> 00:57:13,770
No, it's tricky. This one. Arm it so that.

555
00:57:13,770 --> 00:57:18,590
That. It's just on top of that one.

556
00:57:19,250 --> 00:57:24,300
Just a little bit. That's it. Nearly. Really? Yeah.

557
00:57:24,400 --> 00:57:36,500
They just about I don't know how to direct, you know, line up the black strips I guess that better.

558
00:57:38,210 --> 00:57:43,240
That's it. And then slide it up. Excellent.

559
00:57:44,050 --> 00:57:49,000
Very good. They are. Thank you. Yes. Just.

560
00:57:53,160 --> 00:58:00,510
My father made a model just like this, which was in the article that we wrote in the British Journal of Psychology,

561
00:58:00,510 --> 00:58:09,690
which we sent to Asher and which stimulated him to make his own very, very remarkable picture with the monks going round and round.

562
00:58:10,170 --> 00:58:14,819
Now, I want to say something about these dogs now.

563
00:58:14,820 --> 00:58:27,950
Now, how do we get. Can we get that thing to work? So this was something about these pictures was something that intrigued us.

564
00:58:28,850 --> 00:58:32,030
When you use perspective, here we go.

565
00:58:32,270 --> 00:58:39,080
Yes. When you use perspective, there is a feature which she never actually used.

566
00:58:39,590 --> 00:58:43,730
And here you can see it lined up. I think we've got it just right here.

567
00:58:45,650 --> 00:58:49,690
You see the dogs? Where should we start?

568
00:58:49,970 --> 00:58:53,630
The front here. You see, you have what looks like a little dog and a big dog.

569
00:58:54,020 --> 00:59:01,370
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.

570
00:59:01,700 --> 00:59:05,149
And then this one just just behind that one there.

571
00:59:05,150 --> 00:59:11,210
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.

572
00:59:11,330 --> 00:59:16,040
But when they get to the front, you see they're not. And there's no break in the front at all.

573
00:59:17,000 --> 00:59:22,070
And whereas the break where you see the range is, oops, what's happened?

574
00:59:23,210 --> 00:59:26,390
All right. All right, then. Now we know where the break is.

575
00:59:27,190 --> 00:59:35,390
One going to. Anybody who's inside there is it?

576
00:59:36,440 --> 00:59:38,600
So we took it. Oh, you've got a bat? Yes.

577
00:59:39,920 --> 00:59:46,580
You see, the idea is the break is disguised by this little fellow having his back legs glued onto this piece here.

578
00:59:47,030 --> 00:59:52,040
So it looks at, though, front legs on this part over here where they're not just hanging into space.

579
00:59:52,280 --> 00:59:56,480
So the poor dog would fall over if he didn't have his loose back legs moved.

580
00:59:56,780 --> 01:00:00,140
But you see, that perspective is locally completely correct.

581
01:00:00,500 --> 01:00:08,390
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.

582
01:00:08,720 --> 01:00:13,490
It's just that globally it doesn't make sense or it makes a certain kind of sense.

583
01:00:13,490 --> 01:00:19,970
But it's not something you could construct out of wood without having a brake somewhere or bending it or something like that.

584
01:00:20,390 --> 01:00:27,560
So here, because that dog is in fact a lot further away from the eye from that dog there.

585
01:00:28,370 --> 01:00:33,560
You can have this dog physically bigger than this one, and they look as though they're comparable size.

586
01:00:34,010 --> 01:00:41,170
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.

587
01:00:41,180 --> 01:00:46,250
You can see quite clearly there's no break in the staircase. So it's a very nice illusion.

588
01:00:46,970 --> 01:00:53,990
And I'm sure Escher would have done things if it brought the perspective angle in which for some reason he never actually did.

589
01:00:55,610 --> 01:01:03,560
I want to show you an example. If I now go back to the back to the power point, we can do that.

590
01:01:07,830 --> 01:01:13,000
I'm not sure if it's the next picture, but it should be. That's right. Yes, actually, that's.

591
01:01:13,110 --> 01:01:23,600
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.

592
01:01:23,820 --> 01:01:32,940
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.

593
01:01:32,940 --> 01:01:37,860
So certainly things like this, it wasn't exactly the trend because it's a lot of squares, but it's the same idea.

594
01:01:38,580 --> 01:01:41,790
So this is a thing which was done by Oscar Isaac, right?

595
01:01:41,790 --> 01:01:45,330
His father. And he did a lot of other things, including various staircases and things like that.

596
01:01:46,680 --> 01:01:50,250
So it's worth, I think, pointing out that ideas were explored earlier.

597
01:01:51,540 --> 01:01:55,050
Now, do I do it on this machine?

598
01:01:56,250 --> 01:01:59,700
No, I don't. Oh, here's the gadget over here.

599
01:02:00,150 --> 01:02:03,300
That's it. Is that right?

600
01:02:06,090 --> 01:02:09,180
Oh, this is just a send. Same thing, but done more slick.

601
01:02:09,330 --> 01:02:16,110
You see, that was his, I think his original sketch. And then you can make it look much slicker with modern technology.

602
01:02:16,590 --> 01:02:23,730
But let me show you something else. You see that wasn't by any means the first example of an impossible object.

603
01:02:24,300 --> 01:02:34,320
Yeah, we have a very beautiful picture which was done by Pieter Bruegel, the elder, I suppose, in the 16th century.

604
01:02:34,800 --> 01:02:41,920
And you see in the middle it's the name of the picture is the, is the magpie and the gallops,

605
01:02:41,940 --> 01:02:46,080
and there's a magpie sitting on the top of the gallows, the gallows in the middle of the picture.

606
01:02:46,710 --> 01:02:52,290
And you see, it is an impossible structure because of where it's joined up at the top is

607
01:02:52,290 --> 01:02:59,580
different from the way the to I I think this point to is if we have these two

608
01:03:00,150 --> 01:03:07,290
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.

609
01:03:07,650 --> 01:03:14,340
So it is an example of an impossible object. And if you go way back, I think there are some even earlier ones,

610
01:03:14,340 --> 01:03:21,120
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.

611
01:03:21,130 --> 01:03:24,300
I think people look at this and say, a Brueghel made a mistake in this picture.

612
01:03:24,300 --> 01:03:26,850
Well, that's, of course, ridiculous. He knew what he was doing.

613
01:03:27,300 --> 01:03:33,090
And I think he wants to create some kind of eerie feeling about it, which you couldn't quite put your finger on.

614
01:03:33,090 --> 01:03:34,950
I think that was the sort of he was trying to do.

615
01:03:36,150 --> 01:03:46,440
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.

616
01:03:47,190 --> 01:03:50,190
About the size. It's the dog paradox. You see the dogs.

617
01:03:50,970 --> 01:03:57,240
If you try to do it with perspective, then you have this issue of the sizes as you go round.

618
01:03:57,240 --> 01:04:02,940
The sizes are not consistent. So I was deliberately doing that here in a very extreme form.

619
01:04:03,210 --> 01:04:10,890
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.

620
01:04:11,220 --> 01:04:14,580
But then there's little, little toy creatures walking up there,

621
01:04:14,910 --> 01:04:21,390
and then they climb up the staircase and they become the same size as the children playing at the top.

622
01:04:21,870 --> 01:04:27,770
So there's this inconsistency arising from this size problem.

623
01:04:27,780 --> 01:04:34,860
The size illusion comes about when you combine perspective with the impossible structure.

624
01:04:35,430 --> 01:04:44,400
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.

625
01:04:45,480 --> 01:04:47,490
Unfortunately, I've forgotten who it was,

626
01:04:47,490 --> 01:04:54,690
but I've talked to a mathematician and I was pointing out that these things are illustrations of what mathematicians call comb ology.

627
01:04:55,950 --> 01:04:58,620
And I'm not going to explain what cosmology is, but roughly speaking,

628
01:04:59,070 --> 01:05:03,330
it's the idea that you can have something with a local structure which has a certain ambiguity.

629
01:05:03,630 --> 01:05:09,060
And the ambiguity in the picture is you don't know how far away the object is that you're looking at.

630
01:05:09,060 --> 01:05:13,910
So you might be looking at a very small thing, close up or a much larger thing a long way away.

631
01:05:14,460 --> 01:05:18,450
And that freedom in the picture is something you can't get rid of.

632
01:05:19,110 --> 01:05:23,430
But the fact that you've got that freedom, you can use it in this inconsistent way.

633
01:05:23,940 --> 01:05:30,150
And that inconsistency, the measure of this inconsistency is what mathematicians call homology.

634
01:05:30,660 --> 01:05:34,500
And this American mathematicians. What can you do that from the other group you say?

635
01:05:34,830 --> 01:05:42,900
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,

636
01:05:43,410 --> 01:05:47,010
but then when I thought about it a bit more, I thought, yes, you can.

637
01:05:47,640 --> 01:05:51,930
And so I wanted to use something that Escher used here.

638
01:05:52,230 --> 01:05:57,780
We had this picture before in the film, but he used the very striking effect.

639
01:05:57,780 --> 01:06:04,800
I think you can see best what I mean if you hold your hand up and cover vertically half of the picture.

640
01:06:05,910 --> 01:06:11,040
Now if you look at the left hand picture, you have a certain interpretation,

641
01:06:11,040 --> 01:06:17,129
you've got a chap climbing up stairs and then you go through that thing and then there's somebody kind of dozing,

642
01:06:17,130 --> 01:06:24,690
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.

643
01:06:25,560 --> 01:06:31,620
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.

644
01:06:32,130 --> 01:06:35,010
And this chap is climbing up the stairs and then he goes inside.

645
01:06:35,400 --> 01:06:43,620
And this thing is a kind of a well covering of a lamp or something like that, and it's completely different.

646
01:06:43,620 --> 01:06:46,710
You see, it's, it's, it's, it's the other way round.

647
01:06:47,820 --> 01:06:56,190
But what Escher has done is he's used the ambiguity of which way around it is, it's called concave and convex.

648
01:06:56,190 --> 01:06:59,910
And which way around is it? You see, there's this freedom. It could be one way or the other.

649
01:07:00,390 --> 01:07:04,710
And on one side of. Any one of those is consistent on the other side.

650
01:07:04,770 --> 01:07:09,090
The other is consistent, but you have to have the strip in the middle where it's ambiguous.

651
01:07:09,510 --> 01:07:14,370
And that ambiguity ambiguity allows you to to do this impossible thing here.

652
01:07:14,970 --> 01:07:19,590
But I thought that there was another thing you could do, which is in some ways a bit more subtle.

653
01:07:20,270 --> 01:07:23,520
Although my picture has no comparison, of course, with that one.

654
01:07:23,820 --> 01:07:27,360
But let me just show you. This was a picture I came up with.

655
01:07:27,390 --> 01:07:31,080
I had them on a simple one at a conference, which I showed the next day.

656
01:07:31,460 --> 01:07:36,060
Now, this was an example of something which illustrates the comb ology.

657
01:07:37,710 --> 01:07:42,360
Now, you see, I don't quite know how to describe this because it flips one way or the other a bit too easily.

658
01:07:42,570 --> 01:07:47,850
But if you try to be consistent. Suppose on the right hand side, that shaded part is the floor.

659
01:07:48,270 --> 01:07:53,579
Then you can walk up the stairs and you can go to the left and walk upstairs, then walk upstairs.

660
01:07:53,580 --> 01:07:57,719
Welcome, sir. Then walk onto the top and then walk down stairs.

661
01:07:57,720 --> 01:08:06,900
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.

662
01:08:08,100 --> 01:08:11,970
You see, the flip is. Is a global thing.

663
01:08:12,130 --> 01:08:17,910
You noticed I probably need that trick that we had in the previous talk to feel what went on in the middle.

664
01:08:18,150 --> 01:08:24,030
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.

665
01:08:24,180 --> 01:08:32,159
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.

666
01:08:32,160 --> 01:08:36,720
So I suppose I can get off of that. But you see, you do have an inconsistency.

667
01:08:36,870 --> 01:08:38,280
Inconsistency in the picture.

668
01:08:38,970 --> 01:08:48,270
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.

669
01:08:48,660 --> 01:08:52,800
And I showed him this picture and I said anything wrong, you know?

670
01:08:54,000 --> 01:09:01,620
Sure. Could you could you make that out of wood so you're sure follow your way all the way around and it flips.

671
01:09:01,860 --> 01:09:05,040
Took him a long time to realise that that's an impossible object.

672
01:09:05,760 --> 01:09:14,070
And so there's a certain subtlety in that, which I suppose the, the convex and concave aspect of it flips too easily.

673
01:09:14,430 --> 01:09:19,800
And when you're walking your way around it jumps to the other and you don't realise it's done that.

674
01:09:20,490 --> 01:09:26,729
But if you absolutely consistently all go all the way around, you find that you have an inconsistency.

675
01:09:26,730 --> 01:09:34,060
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.

676
01:09:34,200 --> 01:09:37,320
Well, there's another one. You see, this one has five.

677
01:09:38,160 --> 01:09:45,390
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.

678
01:09:45,840 --> 01:09:49,350
And then this one is another one which is using seven now.

679
01:09:50,100 --> 01:09:57,690
And the shading was always a problem. You have to be very careful about how you do the shading that that that works.

680
01:09:57,690 --> 01:10:10,229
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,

681
01:10:10,230 --> 01:10:11,070
and I'm sure he would have.

682
01:10:12,240 --> 01:10:20,610
This is this is an issue, a well-known Escher picture, and you will see is involves actually some combination of the restricts.

683
01:10:20,850 --> 01:10:28,770
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.

684
01:10:29,010 --> 01:10:36,479
But if you follow them consistently, they're inconsistent with what the other ones do when you in fact,

685
01:10:36,480 --> 01:10:41,910
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.

686
01:10:42,210 --> 01:10:47,730
But when you when they're sort of in the middle, they could point one way or the other and he's done it.

687
01:10:47,730 --> 01:10:48,690
So it's inconsistent.

688
01:10:49,320 --> 01:10:54,660
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.

689
01:10:55,020 --> 01:11:03,150
And it's a very clever you see is it's you can the ones I showed you we're just using straightforward

690
01:11:03,990 --> 01:11:09,270
flat flat surfaces were more or less flat and they're just sort of staircase things.

691
01:11:09,660 --> 01:11:13,530
Whereas here you have a much greater subtlety in the shading on these little bubble things.

692
01:11:13,920 --> 01:11:15,329
And Escher would have, I'm sure,

693
01:11:15,330 --> 01:11:22,830
made of great use of that sort of thing to create impossible structures of kinds that he never have seen and never will see,

694
01:11:22,830 --> 01:11:28,680
except that I suppose other people can use these ideas to produce impossible structures.

695
01:11:29,790 --> 01:11:30,510
Thank you very much.