M.C. Escher

27:44And you get people like Coxeter, Donald Coxeter, another mathematician we might talk about more presently, but he made connections in the mathematical world and word spread. And so he becomes this figure and he talks with mathematicians and they have these conversations and back and forth, which is a lovely thing to see. It's not just one-sided. He's not just learning from them, they're learning from him too. And it all starts in 1954. And so he's corresponding with these mathematicians. Yeah. So for me, I think favourite relationship that he has with a mathematician is the conversation with Donald Coxeter that goes on for many years.

28:22Because Donald Coxeter is, I guess, is it fair to say he's the best geometry of the 20th century? We can argue about that. No, I'm not sure. Yes. Put you on the spot there. Geometry warfare. I know. I'm going to hold the ring here. I have no dog in this particular fight. But yeah, so Donald Coxeter was... One of the best geometrists of the 20th century, I think we'll say. One of the best geometrists of the 20th century. And he was absolutely fascinated by symmetry, and a lot of his work is on symmetry. And he bought actually a couple of Escher's prints at this 1954 exhibition. His wife was Dutch, so could talk with Escher in his own language.

29:03And a couple of years afterwards, after the 1954 exhibition, Then Coxeter was writing a mathematical paper about symmetry and he wanted a couple of illustrations. He wrote to Escher to say, please, may I use two of your tiling designs as illustrations of symmetry in this paper? So Escher says, yes, very nice, very flattered. Start of 1958, Coxeter sends the finished paper to Escher just as, you know, thank you very much. Here's the paper. And Escher has a flick through it. He's not really understanding much of the mathematics, but he sees this amazing picture. and it's a tiling that Escher has never seen before which is drawn within a circle and it's got triangles that tile together but they get smaller and smaller as they get closer to the edge of the circle and because of that it means you can fit kind of an infinite design within this circle and one piece of paper but it's a tiling Escher's never seen so he gets super excited by this he writes to Coxta and says tell me all about this and they had this conversation and as a result Escher produces many beautiful works including his famous Circle Limit series.

30:12Marcus Sotoy, we've already heard about Euclidean geometry, spherical geometry. Now there's... Grace yourself. Yes, exactly. Hyperbolic geometry as well. Explain, you know, in ten easy sentences, explain what that is. Yeah, well these were discoveries of new geometries in the 19th century where strange things happen that don't happen in Euclidean geometry. We've already heard from Sarah about triangles on spheres whose angles add up to more than 180. But these hyperbolic geometries, which are a bit like, if you think of a Pringle Crisp, for example, it curves up one way and down the other. Or if you've gone to the Olympic Park in London and seen Zaha Hadid's aquatic centre, That's another example of a kind of curved geometry which curves up one way and down the other.

31:06And if you draw triangles on the sides of these shapes, the angles add up to less than 180. So these are what we call negatively curved in contrast to the sphere, which is positively curved. So very often these geometries are kind of infinite and we think maybe our universe is perhaps an example of hyperbolic geometry. But what was beautiful was this picture that Escher saw is a realization of this geometry that Poincaré, a mathematician, French mathematician, came up with, which is actually you can capture this infinite geometry within the confines of just a circle. And Sarah already hinted at the idea that triangles kind of get smaller and smaller to the edge of this circle.

31:51Actually, what happens if you take a ruler and you move it towards the edge, the ruler gets shorter and shorter and shorter and shorter. Not if you're in the geometry. It's still a metre length ruler. But if you're outside the geometry, you see that you can fit actually infinitely many metre rulers. They're just getting smaller and smaller and smaller and they never quite reach the edge. So Vresja and the artists in the Alhambra would have just been blown away by this because they're all excited by the way that Tylan can capture infinity. Yet suddenly he's got infinity captured in a finite circle.

32:27And that just sends him on all these just wonderful images that he makes with angels and devils being, instead of just triangles, being tiled to the edges. And he knows, you know, how far can he go? So, I mean, that's a real kind of challenge. But he wants to try and fit as many in as possible to give that illusion. I think I understand what you're getting at. But let me ask Judith Kade. Relativity is probably one of the most famous of Escher's images. Can you describe it for us and why it's so special? Yes, relativity is one of those impossible buildings that you can't really get your mind around.

33:08What you see on the image, it's an impossible building with a lot of staircases. And at first glance, it seems okay, you know, can exist. We see some doors, some archways, seems fitting. And then all of a sudden, the penny drops. There is no gravity in this print. Or it doesn't drop. It was very good. I like that. Or it drops in three different ways. Yeah, exactly. If you turn the print around also, it really doesn't go anywhere, to be honest. No, it's a really funny print in that sense, because people are walking in that print and living in that print like nothing is going on. Nothing is happening.

33:48That's a very normal reality to them. People are going on a romantic walk. They're reading a book and just walking the staircase sideways as if that's just a normal reality. And I think a lot of people really love this image and it's everywhere in popular culture, from like Squid Game, the Netflix series, to Harry Potter. I think it took inspiration from those staircases. So it's still a very well-loved prince. And also the people he has in relativity and in other prints as well, they're kind of weirdly robotic human. They're odd. Yeah. What are they? Who are they? Well, I quite like that about those prints, actually.

34:31I think sometimes it was a bit of a cheat code for Escher, too, because he struggled with human anatomy sometimes. But I think Escher liked them to be anonymous, not to have very clear facial expressions. They are just cohabiting in this space that he created. And that is just very normal. and sometimes he has someone that is maybe looking towards something that he created. So, for instance, in relativity, you see someone looking over the ledge and he's just casually looking, you know. It's just another day at the office. Marcus, relativity also caught the eye of Roger Penrose. What happened with that?

35:15Yes, Roger, a colleague of mine in Oxford, I mean, he was just a student at that time, went to the International Congress and he saw a catalogue. It was just like, what is that going on there? And he suddenly sees these kind of paradoxical worlds. And it's what inspires him to create other versions of these kind of visual paradoxes. And he comes up with this impossible triangle, which is a, it looks like you should be able to build it. He draws it on a piece of paper. It seems to have kind of joins at right angles. But if you follow the whole thing round, you know, locally it makes sense in little parts.

35:52But globally, it just, you can't build it. And it sent Penrose off on this kind of idea of just creating these kind of weird things that sort of you can draw in two dimensions, but can never exist in three dimensions. he sends this to Escher and there starts to be a really good conversation going between them and Escher thinks oh that's really beautiful but it's it's kind of rather abstract and mathematical and he creates out of that the idea of the ascending descending staircase which I think is another of the images which is very famous of Escher's which is this staircase which seems to be climbing up and up and up and there are people walking up they look a bit like monks I'm not sure whether they are monks but uh um but then you look and you see well actually but the staircase comes back around to the beginning again but they've been going up all the time so shouldn't they be on the floor above and and they're not and there are some people going down they're descending and so using this illusion you can create this incredible uh print where it seems to be these staircases ascending but it never gets anywhere sarah one of the extraordinary things about this.

36:58I mean, you've talked about his relationship with Coxeter, and Marcus has talked about his relationship with Roger Penrose. And yet he didn't really have the language of mathematics. So how did this artist and these mathematicians communicate with each other? Was it entirely visual? It was heavily visual, I will say. And there's another mathematician he corresponded with called Polya, who was one of the first people to classify these 17 symmetry groups. And yes, Escher absolutely was all about the diagrams. And he would try and work out how things were put together using his intuitive geometry, you know, geometrical constructions with compass and straight edge.

37:42He always said, I don't understand equations, I don't understand formulas, and therefore thought he wasn't a mathematician. Of course, I disagree with him. But yeah, he would ask and have communications with these mathematicians. Sometimes they would tell him the equations, but he wouldn't really pay attention. But he loved the pictures. So he had a picture that came from a paper by Polya with each of these 17 patterns. And I think what the glorious thing is, you know, as Marcus has hinted to already that, you know, what Escher is doing, he's taking the mathematical idea, the bare bones of a structure.

38:17and then he is adding his artistic interpretation to it. You know, you take a symmetry design that Polya has sketched, which has, I don't know, parallelograms or something, and then Escher creates from it this amazing design with lizards crawling around a piece of paper that's just charming and beautiful, but still incredibly symmetrical. And he does this with the inspiration he gets from Penrose, he does it with the inspiration from Coxeter. You know, Coxeter's picture just has black and white triangles, but then Escher's got fish swimming along beautiful arc lines in hyperbolic space. And Judith, he seems to have this very warm and intense relationship with mathematicians but he didn't have that relationship with everyone.

39:01Can you tell us the story about Mick Jagger and M.C. Escher? Yes, Mick Jagger. Well, Mick Jagger wanted to use one of Escher's images, Verbum from the Second World War and he wanted to use that image on one of his LPs for the Rolling Stones. So you would kind of think, OK, that's quite an honour, right? Not for Escher. He didn't really like that popular kind of music. And he received a letter saying, Dear Maurits, can we please use one of these images? And Escher thought it was an inappropriate question to ask that way. So he didn't allow the Rolling Stones for the image to be used. And he didn't like having his first name.

39:37Exactly. It was Dear Mr. Escher to him. That's brilliant. OK, and now we're moving on to the really weird bit, Marcus, and that is strange loops and consciousness. Yes. How does Escher come into the picture of strange loops and consciousness? Well, people might know of a very sort of iconic book called Gödel, Escher, Bach by Douglas Hofstadter. This is a book actually about consciousness and people probably know who Bach is. And interestingly, Escher loved listening to Bach when he did his work. So already a connection there. But Gödel is the mathematician here, a logician who proves a theorem about the limitations of what mathematics can prove.

40:22But the point is he uses self-reference to make that proof. And Hofstadter got very excited by the idea of mathematics being able to talk about itself. And he coined this phrase a strange loop, something that somehow is a hierarchy. When you get to the top of the hierarchy, you actually got back to the beginning again. And he wrote this book, which is about girdle and consciousness. And his dad said, oh, but you've not got any pictures. And so he said, oh, yeah, what's a good picture of a strange loop? And he knew about Escher's work. And Escher's paradoxical images are wonderful examples of strange loops.

40:58We've already heard the staircase, which you don't quite know. You think you're ascending and then you come back to the beginning. But I think the one image for me which really captures the idea of a strange loop is the hand drawing the hand drawing the hand. There are two images of hands and you look and one seems to be drawing the other, but then you look and it's actually drawing the first one. So this is a very good example of a strange loop where you don't know where the bottom and the top of the hierarchy is. It sort of goes around in a circle. And Hofstadter thought that this is what the brain does.

41:32This is how we achieve consciousness, that we are able to think about our thoughts. It seems to be a great example of self-reference. So Hofstetter thought, all right, this idea of mathematics being able to talk about itself might be the right language to understand how consciousness works. And the illustrations of that were these wonderful pictures of Escher. so I think Marcus has gone some way Sarah to explain why mathematicians like Escher so much what do you think he still remains incredibly popular amongst scientists and mathematicians why is that more so than any other artist I'd say so I think there's a couple of reasons one is it's so great when you know you have an artist that has this interaction with mathematicians and we all love those stories.

42:25In fact, when Escher was creating these beautiful circle limit pictures, he used to say, I'm just going to do some coxetering now, which is maybe the only verbing of a mathematician's name in the service of art that I know of. So we love that, talking to artists, and we believe, mathematicians, that what we do is itself a creative art. We have some of the same motivations, making and exploring beautiful things and ideas. it might be more abstract than some concrete kinds of art but it's it's all the pursuit of the same kind of thing and our love and joy in structure and pattern shared by actually I think all human beings we all love patterns and structures that's why we like the music of Bach and we like beautiful tilings in our homes you know it's not just Escher we all have tiling tiled patterns probably somewhere in our home you know but we are perhaps mathematicians maybe more acutely aware of seeing these patterns and structures everywhere around us.

43:20And so we identify with Escher and, you know, his aesthetic joys are the same as ours. But, Judith, is there a danger here that he's so engaged with structures and patterns that he's missing the emotional aspect of art? Because when you see an Escher, you don't immediately think, oh, my God, that's beautiful. You think that's fascinating. What about that emotional impact? Is it there? Well, does it need to be emotional? I think that's also a question that Escher would ask himself too. I think Escher is portrayed as more rational. He has sort of a scientific approach to art that is maybe different from a lot of other artists, but he has something special in that, in my opinion, too.

44:02What I saw a lot in Asher in the Palace is that people would just come and look at Asher's art with grandparents and their grandkids, people on their first dates, and they just look together and they talk about it. They tell you what they're seeing. And in that sense, they don't feel burdened by, oh, I need to have our historical background to understand it. I need to have read up on Asher. And I think that's actually a strength rather than a weakness. And why is it that he was so averse to being categorised? Well, I think he saw himself firstly as a printmaker and his craft was the most important and his ideas kind of came second to a degree.

44:45And I think he just didn't want to be in this box of, oh, I'm part of this movement. He just did whatever he wanted to do. That's how he would phrase it himself. Sarah, you wanted to come in there. Well, yeah, you asked about why isn't there the emotion that's generated? And for me, there is an emotion that's generated and it's joy because it's the experience of the sublime that you get. If you're listening to a beautiful piece of music by Mozart, it's just this is exactly perfect. This is how nature is. This is how the world is. It's lovely perfection. And you look at an Escher picture and it's just absolutely brilliant, perfectly there.

45:23And that makes me very happy. So I think that's a valid emotion to feel when you're looking at art. Final question for all of you. What's his legacy today? Where do we see Escher? I think this idea that he didn't want to be put in a box is really important because I think we silo our subjects so much. And Escher is a beautiful example of the bridges that you can create between the sciences, mathematics, nature, art, music. I feel that's his legacy. He brought together subjects that traditionally have been seen to be very separate. Anyone else on Escher's legacy, the import? Well, I think his visual language is so recognisable and still decades later.

46:03So I think it will stay with us for many centuries to come. It's the black and white, it's the impossible staircases, it's the mathematics that are underlying his works. It's one of a kind. Sarah? I think there are so many iconic images that are still pervasive in our culture decades and decades after he died and they're going to stay with us. So Escher will always be a part of our legacy, I think. My thanks to Marcus Dusotoy, Sarah Hart and Judith Cadet. Next week, how a million and a half people from India were transported to sugar plantations from 1834 onwards to replace enslaved labourers.

46:42That's Indian indentured labour. Thank you for listening. And the In Our Time podcast gets some extra time now with a few minutes of bonus material from Misha and his guests. What did we miss out? There was one thing I was hoping to slip in, which is... Now's your chance. Yeah, exactly. Which is, you know, very often it feels like a one-way traffic that mathematics inspiring artists. But what's so exciting is seeing when an artist asks a new question which stimulates mathematics in a new way. So Anesha does this because he's absolutely fascinated in colour, not just the shapes, but the colours the shapes have.

47:23And then he's asking, well, what if you want to preserve the color? So, you know, a black tile has to go to a black tile. And this question, actually, mathematicians hadn't thought about. I mean, I said there were 17 different symmetrical games you can play with no color. But if you introduce color, then there's a new theorem which gets proved, kind of inspired by Escher, which is there are, I think, 63 symmetries if you're actually trying to preserve color as well. So I think that's a beautiful example of it isn't one-way traffic. The artist can ask a new question, well, should he matter? Oh, I never thought of putting colour on these things and keeping that invariant.

48:02Sarah, is there anything that you'd like to highlight that we missed out? Well, I think there's a really curious little story about Circle Limit 3, which is one of these circle pictures which is drawn, it's kind of like a map of hyperbolic space. So if we drew a map of the world, we know that we're going to distort some things. And that's exactly what happens with this hyperbolic representation. It's a map. And so some things look distorted. But if we had our hyperbolic glasses on, everything would be fine. But Escher produces this picture. And it looks like what you see in the images, kind of fish swimming along these curved white lines.

48:40They're arcs of circles. And it's a beautiful design, very, very intricate. Took months to make. And he complained to his son George about this. Like it's, oh, you've got these five different bits of wood and they've got four different colours and all of this. It's a nightmare to make. But when Coxeter saw this picture, he saw these kind of lines that the fish were swimming along, these circle arcs, and he thought that Escher had got something wrong about the geometry because there's a rule about the way that these lines have to behave in the hyphonic world, which they have to meet the edge of the circle at right angles.

49:14And these white lines don't do that. It's more like 80 degrees. So he kind of thought artistic license, something's fine. Then it turned out, no, these particular lines, they're not kind of straight lines in the hyperbolic world, but they're lines that are equidistant from each other. So it's a slightly different equation, but Escher had nailed it intuitively. He'd managed to make this. And what it does is it creates within the hyperbolic pattern, it sort of gives you the illusion of a Euclidean one. So it's this really clever way of kind of having a Euclidean thing happening inside this weird hyperbolic world that Coxeter said gave him an entirely new understanding of this kind of geometry.

49:59and he wrote several mathematical papers about Escher's work. And so it's just, it's such a marvellous thing that his first look at it thought, oh, well, it's just, he hasn't quite got that right. And then he realised, no, he's got it, he's got it totally right, but he's doing something different than I thought. Marcus, now, this is fascinating. The Murbia's Strip. I want you to tell us about the Murbia's Strip. Yes, the ants. Exactly, Escher, the ants. Perhaps, Judith, you can describe the ants first of all, and then, Marcus, I want you to tell your amazing story about the Bach-Murbius strip, which just left me completely...

50:38It blew my mind. Judith, tell us about his Murbius strip phase. Yes, well, the Murbius strip, there's one print that he's quite famous for, and the Murbius strip is such a simple mathematical concept. Even I, as a humanities student, can kind of grasp it, I feel. and it's just one sheet of paper that is turned or is twisted and is put together. So it creates this endless loop. And Escher made a very simple print with that in which he has ants walking alongside them, both on the inside and on the outside and on the inside and on the outside. In fact, there is no inside and outside. That's the extraordinary thing about...

51:16It's quite a good example of one of these strange loops, actually, because you think there's an outside and an inside, but when you go around it, you realise that if you try to colour it, this shape, You know, if you didn't do the twist, you could have a red on the outside and blue on the inside. But if you tried to do it with a Mervius strip, the red would go back and you'd find you'd covered the whole shape. So it's actually only got one side to it, which is if you try to cut it in half around the actual loop, it would not fall into two pieces, but would be just one. You know, your listeners have got to try this because it's just such blew my mind when I first saw it.

51:52But yeah, as you say, OK, it's a physical thing, but how does it come up in music? And because Escher was so fond of Bach and you can see that you can feel the connections between Escher and Bach. Tell us about this. This is a great example, again, of artists being drawn to structures that they don't even know are pieces of mathematics. And actually, the Mervius strip hadn't even been discovered yet Bach had already realized it in musical form. So he used to like to create these kind of puzzle canons. So he'd write just a single line of music and then there would be kind of code. So you'd understand, OK, that's the right hand is playing that piece of music, but the left hand is going to do something different.

52:33It's based on that little bit seed that he's written. But the kind of symbols around it say, I'm going to do something. Maybe I'm going to start a little bit later, like a cannon. so this it's actually these puzzle canons that were discovered quite recently that were written after the goldberg variations and one of these canons if you realize it actually what Bach is asking you to do is to take this little seed put it on a merbius strip a transparent one and basically you play with one hand going around the merbius strip but on the other side and everything's reversed because you're seeing it through transparent and and so so the two melodies are going forwards and backwards yes but they are perfectly aligned and they're captured by the music on this merbius strip now bach didn't know that but you know we realized that what he was asking the pianist to do is actually realized by this kind of rather extraordinary geometry So where do we go with Escher now?

53:34If you look at Alhambra from the 14th century and you don't get that much tessellation tiling in art until Escher comes along, is he now part of every artist's education? Do you have to know about Escher if you want to be a serious artist? Well, I think most serious artists are really snobby about Escher. Exactly. I think, you know, the control and the predictability of Escher for some artists is actually too controlling and somehow the unexpected is not there and that's somehow often disappointing for, I think, an artist. And, you know, actually, you know, Bach loved pattern, but he loved disrupting patterns.

54:19So the Goldberg Variations is a perfect example. There's so much structure until you get to the last variation, which is a quadlibit, a musical joke, has nothing to do with anything before. But then you suddenly appreciate the structure because when it's broken. So, and Escher, I'm not sure he breaks, he likes things, as Sarah said, the perfection is part of the charm, but sometimes it's too perfect and therefore doesn't perhaps connect with the messy side of our emotional world. I think the issue is, in art history, Escher has been kind of avoided. When I started working for Escher in the Palace, I came from a museum about Piet Mondrian, the modern artist, and it's a very emotional painter to a lot of people.

55:01And I said, oh, I'm going to work with Escher. And people were like, OK, lovely. And they didn't get it. And maybe also myself, when I started, I was just curious to be surprised. And I was also a bit scared maybe about the mathematical or the scientific side of Escher. And I think that there are so many layers to Escher that it's a bit unfortunate. that is just sort of pushed away from the mainstream art historical perspective. But also I think when you look at Escher exhibitions, they're always unbelievably successful. So it shows how popular Escher is, but maybe not with a typical art historical group of people.

55:41But, you know, maybe we can sway them a bit. Sarah, you've written a book about mathematics and Moby Dick. Going on beyond Escher into other areas of art. Yes. What is it about mathematics and literature where, you know, I can see the relationship with music, but the relationship with literature. Explain to me why you think maths and literature live together and react together. So the book is about the connections between mathematics and literature. And really one of those connections is structure and pattern, because that's present in all our forms of creative expression. Poetry, for example, that is full of patterns and understanding those and exploring them is fascinating, but also it provides more structures for writers to work with and use.

56:31So if you understand the patterns of poetry, maybe you write different kinds of poetry. So poetry, but also novels. One of my favourites is The Luminaries, has this wonderful mathematical pattern inside it. But then there's writers who use mathematical ideas in their work. So Herman Melville is one of them. I mean, loads of mathematical ideas in Moby Dick. So this sort of symbolism comes into writing numbers in fairy tales. It all pervades literature. And then, of course, you get the stories that involve mathematicians themselves, like Moriarty. Sherlock Holmes is a mathematician. But as the author of six nonfiction books, I'm not necessarily engaged with mathematics in any way.

57:12You are. You are. Here's why, right? Here's why. It's all geometry, right? OK, when you write anything, you will have letters that make up words, that make up sentences and paragraphs and chapters and volumes. And in geometry, you have points that make lines, that make planes, that make spaces. So there's this hierarchy of dimensions, which is exactly like what's happening in literature. And at every stage, you have a choice about, do I divide my book into chapters? How am I going to shape my paragraphs? How are my sentences fitting together? Mix of long and short words in a sentence, mix of long and short sentences.

57:52And those kinds of decisions, that is you doing mathematics without knowing it. Right. So basically it's two different languages, but they're both performing the same function. Yeah, yeah. And they're translatable, presumably at some level. Yes, exactly. And even there's another kind of link, which is if you are writing something down, describing something about the world, fiction or non-fiction, you are simplifying to get at the greater truths. Tell me about it. Right? So that's exactly what mathematics does. Like there's no such thing in the real world as a truly perfect circle. But by simplifying and saying this is a circle, constant radius, so on, you can get at truths that tell you something about the world in the same way that a piece of factual writing can get at deeper truths by leaving out unimportant details.

58:38So it's, again, it's the sort of same methodology in mathematics and in writing that we're trying to uncover the truth. Well, just as I would recommend anyone to go to an MC Escher exhibition, I would also say that if listeners are anywhere near Alhambra in Granada in Spain, not to be missed. An absolutely magical experience. So thank you very much. And here comes Martha. Tea, coffee. Yay, tea. Tea, please. Can I have a peppermint tea, please? Tea, tea, peppermint tea, coffee. In Our Time with Misha Glennie was produced by Martha Owen. It's a BBC Studios production for Radio 4. The Moral Maze on BBC Radio 4.

59:21I've never been more concerned about the future of humanity than I am now. Examining one of the week's main news stories through an ethical lens. If we don't do something, millions will die, billions will die. That's the state of play here. Sometimes combative, sometimes provocative, always engaging. I'd like to go one level deeper and talk about your fundamental moral commitments. Do you have any? The new series of The Moral Maze with me, Michael Burke, from BBC Radio 4. Listen now on BBC Sounds.