In Our Time - M.C. Escher

Episode Date: May 14, 2026

Misha Glenny and guests discuss the work of Maurits Cornelis Escher (1898-1972), the graphic artist and printmaker best known for his impossible buildings, paradoxical perspectives, and repeating geometric patterns. Born in Leeuwarden and trained as a printmaker, Escher visited the Alhambra in Granada and found inspiration in the tessellating shapes of Islamic art. Through his career he went on to create some of the most famous images of the twentieth century and has been called a one-man art movement. After his work was exhibited in a 1954 conference, Escher’s work also caught the eye of mathematicians who appreciated his intuitive geometric precision. Escher was influenced by their work, and they were influenced by his – despite Escher never thinking he was actually very good at maths himself.   WithMarcus du Sautoy Simonyi Professor for the Public Understanding of Science, Professor of Mathematics and Fellow of New College, University of Oxford   Sarah Hart Professor Emerita of Mathematics and Fellow of Birkbeck, University of London, and Fellow of Gresham College   And   Judith Kadee Exhibitions project manager and public programme curator at Hague Historical Museum   Producer: Martha OwenReading list:Marcus du Sautoy, Blueprints: How Mathematics Shapes Creativity (Fourth Estate, 2025)Marcus du Sautoy, Finding Moonshine: A Mathematician’s Journey Into Symmetry (Harper Perennial, 2009)Bruno Ernst, The Magic Mirror of M.C. Escher (Taschen, 2007)M.C. Escher, M.C. Escher: The Graphic Work (Taschen America Llc, 1992)Miranda Fellows, The Life and Works of Escher (Siena,1996)Frederico Giudiceandrea, Escher op reis or Escher’s Journey (Publisher Wbooks, 2018, in Dutch)Sarah Hart, Once Upon a Prime: The Wondrous Connections Between Mathematics and Literature (Flatiron Books, 2023)Douglas Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid (first published 1979; Basic Books, 1999)Siobhan Roberts, King of Infinite Space: Donald Coxeter, The Man Who Saved Geometry (Profile Books, 2007)Claudio Salsi, Paolo Branca and Claudio Bartocci (eds.), M.C. Escher. Tra arte e scienza. Catalogo della mostra (24 Ore Cultura, 2025, in Italian)Doris Schattschneider, “The Mathematical Side of M.C. Escher” (Notices of the American Mathematical Society, Vol. 57, 6, 2010)Doris Schattschneider, M.C. Escher: Visions of Symmetry (Thames and Hudson Ltd, 2004)Wouter van Reek, Nadir & Zenith in the World of Escher (Leopold, 2019)In Our Time is a BBC Studios productionSpanning history, religion, culture, science and philosophy, In Our Time from BBC Radio 4 is essential listening for the intellectually curious. In each episode, host Misha Glenny and expert guests explore the characters, events and discoveries that have shaped our world.

This BBC podcast is supported by ads outside the UK. In the Range Rover Sport, performance is more than a promise. It's something you feel on every drive. With the choice of powerful mild hybrid and plug-in hybrid engines, Range River Sport responds instantly, bringing unbridled power and precise handling. It's a perfect balance. Explore more atrangerover.ca. When it's time to scale your business, it's time for Shopify.

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The Signal Awards recognize the podcast that define culture, and being honored by the Signal Awards, sets your production team apart with recognition from the industry's top experts. and access proof that your work is a standard bearer for podcasting worldwide. By entering, your work is heard by the Signal Awards Judging Academy, an invitation-only body of podcast professionals from acclaimed organizations, which include the BBC. Grow your audience, celebrate your team, and stand out.

The final entry deadline to submit is the 26th of June. Enter your podcast at signalaward.com for consideration. This is In Our Time from BBC Radio 4, and this is one of more than a thousand episodes you can find in the In Our Time archive. A reading list for this edition can be found in the episode description wherever you're listening. I hope you enjoy the program. Hello, never-ending staircases, dizzying twists of perspectives and illusions.

that seem to defy the laws of physics. That's the world of the Dutch graphic artist and printmaker Maurits Cornelis Escher, known by us as M.C. Escher. Born in 1898, Escher was inspired by the geometric shapes of Islamic art and went on to create some of the most famous images of the 20th century. He's been called a one-man art movement. But Escher has also been celebrated by generations of mathematicians, for his technical, geometric precision,

despite never actually thinking he was very good at maths himself. Well, with me to discuss M.C. Escher, a Marcus Dussotoy, Simonyi Professor for the Public Understanding of Science, Professor of Mathematics and Fellow of New College, University of Oxford, Sarah Hart, Professor Emeriter of Mathematics and Fellow of Birkbeck College,

University of London, and Fellow of Gresham College, and Judith Cadet, exhibition's project manager and public program curator at Hague Historical Museum. And I'll start with you, Judith Kadei. You've been curator of the Escher in the Palace Museum in the Hague. So tell us a bit about Escher. What was his early life like?

And was he always interested in maths? Absolutely not. Escher grew up in the north of the Netherlands. That's where he was born in Leavarda and grew up in Arnhem, the region in the east. And what was special about Escher, I think he had quite a nice upbringing. His father had a very good job. It was one of the first hydraulic engineers to go to Japan. And he had a wealthy upbringing in that sense, loving family.

But at the same time, he was also struggling sometimes a bit because he was quite poorly as a kid. I had a lot of illnesses. And also, he was not good at math. Especially, well, he was not a great student, to be honest. So he didn't pass his final exams. He didn't pass some courses. And he struggled. And I think that's something good to take away from this.

So if he was struggling, did he end up studying anything? And if so, what? Well, at first he dabbled a bit in architecture. That's where his parents wanted him to go into because that would give him a nice and proper job. But that didn't really work out. So he went to a school for both architecture and arts. And actually he switched directions.

So he got scouted by his teacher, Samuel Jeserun Miskita, who said, you shouldn't be making buildings, you should be making prints. So he talked to his parents into making Asher switch as a young man. And what was his artwork like in this period in the 1920s when he was studying? I think it's fairly traditional, I would say. It was printmaking in the early 20s. So he was making still lives, portraits, cities. escapes, biblical studies also.

And it was a few years for Escher just to figure out, how does printmaking work? How do you go about it? And what kind of subjects am I interested in? So it was interesting. Those were formative moments, but probably not the work that he would have become famous with nowadays. So Sarah Hart, Escher traveled to Spain, and this was very formative for him. What did Escher see when he visited Aloysia? the famous Moorish Palace in Granada.

How did that inspire him? So he, yes, he travelled a lot in Italy in Spain, and in the 20s he was perhaps more focused on making landscapes. But his visits to the Alhamra Palace gave him this profound love of symmetrical tiling patterns, which are typical of Moorish art. One of the reasons for that is that there's traditionally a prohibition against representing living things. So if you want something to decorate your walls with,

a beautiful geometric pattern is a good option. So in the Alhambra Palace, there are these wonderful tile work patterns that just have beautiful symmetries, lots of different kinds of symmetries. And Escher was really inspired by these. And in fact, after spending a few days at the Alhambra drawing designs, copying the pictures that he was seeing, his work really took a new direction in 1936. And the Alambra Palace was built largely in the 14th century,

So did the creators of this extraordinary tessellation, this tiling, did they understand mathematics or do we think this is instinctive? They absolutely understood mathematics. They would create very precise diagrams using kind of, you know, geometrical tools, compasses, and you can construct things by circle constructions, hexagons, 12-sided figures, geometrically that, you know, within the limits of what can be produced in the real world, are as perfect as they can. can be. The only perhaps thing that cannot be done in that sense is that if you are trying to

create a pattern that could go on forever, you know, you've only got the finite extent of the wall or the floor that you're making it on. So you can hint at infinity, but you can't say all of infinity in this picture. Now, as I said, this is called tessellation. Can you explain that concept in geometry? Yeah. So tessellation or tiling, imagine a tiled floor that's just, for example, got different colours squares. So black and white squares, tiling a floor. Now, if you've got any sense, you're not going to want to leave gaps and you're not going to want the tiles to overlap each other. So a tiling mathematically is a pattern of shapes, and if we're talking about, you know, a plane, a flat plane, a pattern of shapes that will exactly fit together with no gaps, no overlaps,

and could be extended, theoretically could be extended forever. So squares fit together just perfectly, they don't overlap no gaps, and they could carry on forever, and that would be a square tiling. So we all know this from our bathroom floor. Exactly. But what have we got apart from squares? Well, so if you boil it down to the basic possibilities, you can think of having one kind of tile, which you then use throughout your design. Those tiles, and our favourite kinds of ones as mathematicians, are regular ones.

So like a square, all the sides are the same length, all the angles are the same, Another example would be the regular hexagon that bees love to make their honeycombs out of. So it's got six sides and all the sides are the same length and all of the angles are the same 120 degrees. Another example is an equilateral triangle. So those are regular polygons. They are shapes made of straight lines where all the angles and all the sides are equal. And you can't make these kind of tiling and tessellations with every single possibility because the angles don't work.

So you can take four squares and put them around a point because squares have an angle of 90 degrees and four 90s are 360. Right? So it works. But if you try and do it with something like a pentagon, a regular pentagon, the angle there is 108. And 3008s are less than 360. And four of them are more. And so you'd either get a gap or an overlap. So you cannot make one of these lovely regular tiling patterns with regular pentagons. Okay.

So Marcus de Sertoy, what Sarah is? hinting out there, if I understand it correctly, is that symmetry here is a key. What do we need to know about the mathematics of symmetry? Well, it's very interesting because, I mean, Sarah's right that the artists in the Alhambu were very mathematically adept, but what they didn't have was a language to understand what symmetry really is. And that only came at the beginning of the 19th century with a French mathematician Everez Galois, who began to understand, you know, how do you articulate what's happening on these walls, the symmetry of these walls? And I think most people listening will probably think, oh, symmetry, that's about reflectional symmetry.

And you certainly see that in the square and the hexagon. But there are other sorts of symmetries as well, rotational symmetries. But what Galois got to was to think about these walls and what makes them symmetrical. It's about the way that you can move the tiles, if you lifted them up off the wall, move them around on block and then put them back down again and they would sit perfectly inside the outline that had been left there. So what you're looking for is what are all the different ways that I can move the tiles such that they sit back down again on top of each other. And those are the symmetries of the wall. Now, I think the artists in the Alhambra were exploring what's possible with all

of these different designs. You know, you can take a triangle, but you can put a lovely little wave on the side of the triangle and that destroys the reflections, but you can still rotate it. And you can really see the artists there just exploring what's possible, but what are the limitations? So I think Escher as well was quite excited by, maybe I can make some new designs. But then he gets a letter from his brother who says, you know what, there's actually a mathematics behind all of this. And he was a geologist and a crystallographer. And crystals are somewhere where you see a lot of symmetry. And he said there's actually a limit to the number of possible symmetrical games you can play. And there are actually only 17 underlying symmetrical games

you can play. And he sent a paper with these to Escher and Escher began to understand, ah, okay, so did the artists in the Alhambra discover them all? Can I do different realizations of these 17? And it set him off on this kind of journey of the mix of mathematics and art. So Escher began to understand that mathematics was key to the structure. of some, if not all of his art. Is that something that you can generalise? Is mathematics underlying most art that we see, whether visual art or other types of art?

I would say yes, because if I was going to define mathematics, I would say it's the study of structure. And for an artist, structure is absolutely crucial for their creativity, for their framing of what they're doing. And so I think that what's interesting, you see a huge dialogue across the centuries between mathematicians and artists. I mean, take the Renaissance and Leonardo.

He absolutely captures the fusion of mathematics and art. But I think that what's exciting is an artist seeing new mathematical structures and being an inspiration for new framing. So I think Escher, beginning with his kind of tiling, chose quite classic sort of symmetries. But when he saw the other possibilities with these 17 different design. So he then, it pushed him in new directions to try out, oh, how can I realize this rather exotic kind of symmetries with my tiling? And so I think it became an inspiration for him

to try new things. And for me, I think that's where you see a lot of artists being inspired by seeing what's in the mathematician's cabinet of wonders that I might be able to use to frame what I'm doing. But you did, Cadet, before we get to the mathematics, of it. Let's go back to nature. He lived in Italy for a while, but he leaves in 1935. Why did he leave in 1935? But also, what did he pick up in Italy? Why was Italy so important to him? Asher loved Italy. He lived in Rome for over a decade. He met his wife in Italy. He had two beautiful kids there. And he loved going, well, what we probably nowadays would call backpacking. So who would sometimes bring a donkey also on like this off the beaten track.

And he made a lot of prints and photographs, drawings on those journeys. And what he loved in Italy was you have the high mountains, the low valleys. There's perspective everywhere. And there's contrasts everywhere, dark and light, high and low. And that ended up in his work too. So in 1935, Asher decided to leave with his family because fascism was on the rise and he really didn't like that. And also one of his kids had early signs of tuberculosis. So in that period of time, mountain air was very well recommended.

So the family moved to Switzerland. But he missed it tremendously, Italy. And it sort of stuck with him for decades. And you can see that, can't you, in the buildings that he uses when he's creating those strange images of stairs ascending and descending and that sort of thing. There's still that Italianate sense. Absolutely. Yeah, sometimes it's in the background. So you see a mountain ridge that comes back to, I think, sometimes a prince 30 years later, but also indeed archways or particular kinds of roofs that are very, very recognizable from the Amalfi Coast. So it stays with him throughout his life. And nature as well is very important because for all of his sort of weird shapes and symmetries and so on, he's got a lot of animals there as well.

Yeah, animals, but just nature in general too. So he's always focused on trees and mountains and whatnot. And he would love to go on hikes also in the Netherlands too. So that's always a theme in his work as well. So Marcus, on that issue of nature, what sort of things was he exploring? How does that turn up in his work? Well, I think, you see, we've got this connection between mathematics and art. And I think nature is the thing which binds them together.

because I think we're all responding to structures that we're seeing in the natural world. I mean, Sarah's already referred to the idea of the hexagon, well, that's, we recognize as the beehive. So I think that this actually is a very important component of his work, that he's looking at these connections between artistic expression, the underlying mathematics. But actually, these are all structures that we're seeing in the natural world. But I think this, coming just back to his time in Italy in the Amalfi Coast, I think there's something really interesting about the move in his art because you see it going from very three-dimensional. He does these lovely cascading villages down the side of cliffs.

It's very three-dimensional, yet when he then goes to the Alhambra, which is very two-dimensional, you really see this shift in his art. And there's a wonderful piece of art called metamorphosis, which he almost expresses in that piece on the left-hand side of the piece. It's quite a long sort of horizontal piece. You see the Amalfi Coast, but it gradually changes

until you've just got these two-dimensional kind of Chinese boy figures which are interlocked with each other. He's almost expressing in that piece, I'm moving from my three-dimensional world into just working in two dimensions. Yes, it's kind of a narrative of his own artistic trajectory.

Yeah, it's very knowing in a way. It's kind of interesting. Extraordinary piece of work. Sarah Hart, you've told us about tiling and tessellation. I think I managed to grasp it. But he also dabbled in something called spherical geometry. Now, I want you to be gentle in explaining this because if the listeners are anything like me,

we're beginning to stretch my understanding of mathematics, yeah? Yeah, so Escher actually himself, as has already been said, he did not think he was any good at mathematics. And I'm sure Marcus would agree, and all mathematicians, that that is only because it's a misunderstanding of what mathematics is. Because he was very good at pattern and he was very good at structure and he loved those things and was brilliant at working with them intuitively. That for me makes him a great mathematician. So it's not just all about equations and formulas. And what Escher was doing, you know, as metamorphosis shows, and he said he was moving from landscapes to mindscapes, which is a lovely little phrase. is exploring what is possible in this kind of geometry. So his work where he was making tiling patterns that are on a flat surface, like the tiled floors or walls, there are some limitations, the 17 collections of symmetries that Marcus mentioned. And at some point, he'd explored these at length,

and he was sort of thinking, well, what else might there be? And there's another kind of geometry that, in fact, is all around us, and that is the geometry of the surface of a sphere. We all live on the surface of a sphere. And the geometry on a sphere is a bit different. Example, when you're going on a long haul flight and you go from A to B, you don't go in a straight line as marked on a map. You go in a curved path because that's the shortest path.

So those shortest paths geodesics, they're like the lines on a sphere. And if you use those lines to make shapes, you can get triangles whose angles are up to more than 180 degrees, for example. So if you have different rules of your geometry like we have on the sphere, then you can have different patterns and possibilities. And so Escher explored, for example, in 1942, he makes a spherical version of the angels and devils image that he'd originally made on a plane.

Now he makes it in three dimensions on a sphere, and there's a third version that will come to later in another kind of geometry. So he's exploring again what you can do if you change your situation. There's a wonderful example of that, which I've got to mention, because it's one of my favourite examples,

which is a chocolate box that Escher made. It's an icosahedron, so it's a platonic solid. Anyone plays Dungeons and Dragons. It'll be one of the dice in your box. It's made out of 20 equilateral triangles. So it's almost spherical. It's trying to be spherical. But he made this on this tin box,

which is celebrating the 75th anniversary of a chocolate manufacturer. He put these wonderful examples of starfish and shells. And it's just a beautiful... When I retire, I'm hoping that somebody is going to give me one of these little tin boxes. I think there were about 7,000 made. So they're out there. But it's just a thing of beauty, but a beautiful example of, oh yeah, what have I put these shells and starfish on a three-dimensional shape?

And Yudit, can you tell us a bit about another famous spherical object? And that's the hand with reflecting sphere. Can you describe that to us? It really is one of the weirdest, most. remarkable pieces of art I've ever seen. It's a lithograph Asher made in 1935. We see Asher in the middle of the print

and he's sitting in his apartment in Rome. So it feels very lifelike, right? And you see him holding the sphere that he is reflecting in. And if you look at Escher's hand in the sphere itself, you see him holding it with his right hand. Still makes sense, right?

And then you look at the hand below that's holding the sphere. That's his left hand. So it's already a mirror image of a mirror image. And then also there's another layer. This is a lithograph and a lithograph is a print. So it's always printed a mirror image anyway. So it's a mirror image of the mirror image of the mirror image.

So it's one of my favorites to point out also. But also I gather very accurate in terms of the scale of the objects in the sphere. Absolutely. When I just started working at Escher in the Palace as a curator, we got an email of someone who rebuilt Aschers apartment, and they said it was completely accurate. Only the table on the right, it was tailored just a little bit to fit the image,

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Based on the best-selling novel, The Other Bennett Sister, now streaming only on Britbox. Watch with a free trial at Britbox.com. When it's time to scale your business, it's time for Shopify. Get everything you need to grow the way you want. Like all the way. Stack more sales with the best converting checkout on the planet. Track your cha-chings from every channel, right in one spot,

and turn real-time reporting into big-time opportunities. Take your business to a whole new level. Switch to Shopify. Start your free trial today. The Signal Awards recognize the podcast that define culture, and being honored by the Signal Awards sets your production team apart with recognition from the industry's top experts and access proof that your work is a standard bearer for podcasting worldwide. By entering, your work is heard by the Signal Awards Judging Academy, an invitation-only body of podcast professionals from acclaimed

organizations which include the BBC. Grow your audience, celebrate your team and stand out. The final entry deadline to submit is the 26th of June. Enter your podcast at signalaward.com for consideration. So most of Escher's work wasn't these kind of 3D objects. What kind of techniques and mediums was Escher using at this time? So Escher was mostly a printmaker. So always two-dimensional. So on paper And I like that Marks that you just mentioned sort of this mixture of 3D and 2D because Asher loved playing with

making something like a piece of paper just a three-dimensional image also and Asher was really a craftsman so he loved the woodcutt, he loves lithographs and especially the woodcuts I always enjoy explaining because it's so difficult and technical Escher would have a block of wood he would draw an image on it

and he would gouge the whole block of wood. So he would cut the image out and then he would put some ink over the top layer. Only the top layer would end up on a piece of paper. And it was a procedure that could last days, if not months for his bigger works like metamorphosis too. That took him five months to make all the 20 different wood blocks. And it's also an unforgiving material wood. So if you cut a little bit too much out, you can't get it back. So that was really a struggle. But he really enjoyed. enjoyed that. And there was also a lithograph of which I gave the example of Handwith Reflecting Sphere, which is more like a drawing. So it's a drawing on a lithographic stone

and with a chemical procedure it's put on paper as well. And it's really, Asher loved the printmaking and he loved the craft of it. And sometimes indeed he dabbled and he made a very interesting chocolate tin or something like a tiling pattern. But his basis was really on paper. Sarah, can you tell us why 1954 was a turning point for Escher? Because he's well into his 50s now. He's established to a degree, but he hasn't reached anything like the fame that he has now. So what happens in 1954? Well, there was a big mathematics conference in Amsterdam in 1954.

This is the International Congress of Mathematics, which is held every four years and still is, as I understand. Yes, every four years, and it's at that conference that the Fields Medal is awarded. So, you know, everybody who's everybody, anybody in mathematics comes along with this conference. And it's kind of a jambore you all get together. Thousands of people are there. To go along with this, there are often cultural activities. And that year, the organisers thought, well, we've got this local artist, Escher, who's, you know, mathematicians seem to like his stuff.

So why don't we do an exhibition of his work? So many people, many mathematicians come from all around the world, heard of his work and saw it for the first time. And 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. Because Donald Coxter is a, I guess, is it fair to say he's the best geometry of the 20th century? We can argue about it. No, I'm not sure.

Well, yes. We're on the spot there. Geometry warfare. I know. I'm going to hold the ring here. Yeah. I have no dog in this particular fight. But, yeah, so Donald Coxeter was...

One of the best geometers of the 20th century, I think we'll say. 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 good talk with Escher in his own language. And a couple years afterwards, after 1954 exhibition, 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 a you know thank you very much here's the paper and Escher flas 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 could 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 Coxger 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. Marcus Soto, we've already heard about Euclidean geometry, spherical geometry.

Now there's... Grace yourself. Yes, exactly. Hyperbolic geometry as well. Explain, you know, in 10 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 Zaharadid's, the aquatic centre, that's another example of a kind of curved geometry which curves up one way and down the other. And if you draw triangles on the sides of these shapes,

the angles add up to less than 180s. 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 Poincoré, 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. Actually, 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 meter-length ruler, But if you're outside the geometry, you see that you can fit actually infinitely many meter rulers.

They're just getting smaller and smaller and smaller and they never quite reach the edge. So, Frasier, and the artist in the Alahambra would have just been blown away by this because they're all excited by the way that Tyling can capture infinity. Yet suddenly he's got infinity captured in a finite circle. And that just sends him on 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? 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 Kadei. 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. What 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's very good, John, like that. That is a very fair point. 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 happening. That'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-oh, they're odd Yeah What are they? Who are they? Well, I quite like that about those prints, actually. I think sometimes it was a bit of a cheat code for Escher too because he struggled with human anatomy sometimes. But I think Asher 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 just, but he's just casually looking. You know, it's just another day at the office. Marcus, relatively also caught the eye of Roger Penrose. What happened with that?

Yes, 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. But 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 could 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 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 Eschers, 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 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 they're not.

But there are some people going down. They're descending. And so using this illusion, you can create this incredible print where it seems to be these staircases ascending, but it never gets anywhere. Sarah, one of the extraordinary things about this, I 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 Polyar, 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.

He 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 Pollyar 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, is taking the mathematical idea, the bare bones of a structure, and then he is adding his artistic

interpretation to it, you know, and you take a symmetry design that Pollyer 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, Coxter'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. Can 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, Verben from the Second World War,

and he wanted to use that image on one of his LPs for the Rolling Stones. So you will kind of think, okay, that's quite an honor, right? Not for Escher. He didn't really like that popular kind of music. And he received a letter saying, Dear Mauritz, can we please use one of these images? And Asher 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. Exactly. It was dear Mr. Escher to him. That's brilliant. Yeah. Okay, 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. But 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're not got any pictures. And so he suddenly thought, oh, yeah, what's a good picture of a strange loop? he knew about Escher's work and Escher's paradoxical images of wonderful examples of strange loops. We'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. This 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 Hofstadter 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. action with mathematicians and we all love those stories. In 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, we're 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 absolutely. straight than some concrete kinds of art, but 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 tilinges in our homes. You know, it's not just Escher.

We all have tiled patterns probably somewhere in our homes. You know, but we are perhaps mathematicians maybe more acutely aware of seeing these patterns and structures everywhere around us. And so we identify with Escher and, you know, his aesthetic joys are the same as ours. But, Judith, is there in 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. What I saw a lot on Escher in the palace is that people would just come and look at Escher'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 burdens 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 categorized? 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.

and 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, this lovely perfection and you look at an Escher picture and it's just absolutely brilliant, perfectly there. And 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. 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 recognizable and still decades later. So 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 kinds. 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 actually will always be a part of our legacy, I think. My thanks to Marcus Tussotoy, 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. That'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. What's so exciting is seeing when an artist asks a new question which stimulates mathematics in a new way. So Annesia does this because he's absolutely fascinated in colour, not just the shapes, but the colours the shapes have.

And then he's asking, well, what if you want to preserve the colour? 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 colour. But if you introduce colour, then there's a new theorem which gets proved, kind of inspired by Escher,

which is there are, I think, 63 symmetries if you actually try and preserve colour as well. So I think that's a beautiful example of, it isn't one-way traffic. The artist can ask a new question, which is the math, I'd never thought of putting colour on these things and keeping that invariant.

Sarah, 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, swing. along these curved white lines, their 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 hyperbolic world which they have to meet the edge of the circle at right angles. And these white lines don't do that. It's more like 80 degrees. So he kind of thought artistic licence, 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. And 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 realized, 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 Murbius Strip. I want you to tell us about the Murbius' trip, exactly, Escher, the ants.

Perhaps you did, you can describe the ants first of all, and then, Marcus, I want you to tell you're amazing. story about the Bach-Murbius strip, which just left me completely. It blew my mind. Judith, tell us about his Mubius strip phase. Yes, well, the Mubius strip, there's one print that he's quite famous for, and the Mubius strip is such a simple mathematical concept. Even I, as a humanity 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 Azure 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 on the outside. In fact, there is no inside and outside. That's the extraordinary thing about... It's quite a good example of one of these strange loops, because you think there's an outside and an inside, but when you go around it, you realize that

if you tried 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 Mervis strip, the red would go back and you'd find you'd cover 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, it blew my mind when I first saw it.

But yeah, as you say, okay, 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 the, You can feel the connections between Escher and Bach. Tell us about the... 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 Mubius strip hadn't even been discovered, yet Bach, who 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, okay, the right hand is playing that piece of music. but the left hand is going to do something different. It'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 cannons, 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 round the Mubias strip, but on the other side, and everything's reversed because you're seeing it through transparent. And so... So the two melodies are going forwards and backwards, but they are perfectly aligned. And they're captured by the music on this Mervius strip. Now, Bach didn't know that,

but we realised that what he was asking the pianist to do is actually realised by this kind of rather extraordinary geometry. So where do we go with Escher now? If you look at Al-Hambler 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. 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 actually, you know, Bach loved pattern, but he loved disrupting patterns. So the Goldberg Variations is a perfect example.

There's so much structure until you get to the last variation, which is a quad-libit, 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, Asher has been kind of avoided. When I started working for Escher in the palace, I came from a museum about Pete Mondrian, the modern artist,

and it's a very emotional painter to a lot of people. And I said, oh, I'm going to work with Asher. And people were like, okay, 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 Asher. And I think that there are so many layers to Asher that is 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. But, you know, maybe we can sway them a bit now. Sarah, you've written a book about mathematics and Moby Dick. Going on beyond Escher into other areas of art. 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, there 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. So if you understand the patterns of poetry, maybe you write different kinds of poetry. So poetry but also novels.

One of my favorites 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. Loads of mathematical ideas in Moby Dick. So this sort of symbolism comes into writing numbers in fairy tales. it's all pervades literature. And then, of course, you get the stories that involve mathematicians themselves, like Moriarty.

Sherlock Holmes is arch-knowess is a mathematician. But as the author of six non-fiction books, I'm not necessarily engaged with mathematics in any way. You are, you are. Here's why, right? Here's why it's all geometry, right? Okay, 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. and 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 nonfiction, 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. So 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 and 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, peppermint tea. Coffee. In our time with Misha Gleney was produced by Martha Owen. It's a BBC Studios production for Radio 4.

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