The Logic of Escher's Impossible Worlds

M.C. Escher was a Dutch graphic artist celebrated for merging mathematical principles with visual art to explore the nature of reality. His work extensively utilizes tessellations, non-Euclidean geometries, and fractals to represent the concept of infinity on a flat surface. Throughout his career, he collaborated with prominent mathematicians like Roger Penrose to construct impossible architecture and complex optical illusions. Escher’s creative process was deeply influenced by natural patterns, such as those found in shells and landscapes, alongside the rhythmic structures of Bach’s music. Today, his legacy persists through his profound impact on popular culture, including significant influences on modern cinema and video games. These sources provide a comprehensive look at his artistic evolution, technical mastery of printmaking, and enduring intellectual contributions.

Transcript

Speaker 1  0:00  
Imagine this. You're looking at a piece of paper, and it's tacked to a drawing board. It looks old, maybe

Unknown Speaker  0:06  
a little bit yellowed, just a standard piece of parchment, exactly.

Speaker 1  0:09  
And on this paper, there's a drawing two hands. But here's where it gets weird. You know, this is the paradox that just makes you stop and stare.

Speaker 2  0:17  
It really does. It forces you to lean in because the hands,

Unknown Speaker  0:21  
they aren't flat. They're coming out of the paper. You

Speaker 2  0:24  
see the wrists first, and they look like sketches, just graphite on paper, two dimensional, lifeless.

Unknown Speaker  0:30  
But as you look up, as your eye

Speaker 1  0:31  
travels up the forearm to the fingers, they become real. They look fleshy, three dimensional. They feel like you could touch them. And what are they doing? They're holding pencils, and each hand is sketching the other one into existence, throwing hands.

Speaker 2  0:46  
It's got to be one of the most recognizable images in, well, in the history of art, and definitely one of the most intellectually

Unknown Speaker  0:55  
chewy. I guess you could say chewy.

Speaker 1  0:56  
I like that. It's a lithograph from 1948 and it just perfectly sums up this strange, looping, self referential world of the man we're talking about today.

Speaker 2  1:06  
It's a visual loop. It defies all logic, but at the same time, it feels so right, almost disturbingly correct.

Speaker 1  1:13  
It is. It creates this immediate question in your head, who's the creator and who's the created? Is the left hand drawing the right or is it the other way around? It's a brain breaker from the word go, it really is. And on that note, welcome back to the deep dive. Today. We are tackling a subject that I have to say I've been excited about for a very long time, because it sits right at that intersection that we love so much, that place where art, math and psychology all crash into each other.

Speaker 2  1:41  
We are, of course, talking about the work of MC Escher,

Unknown Speaker  1:44  
Moritz Cornelis. Escher,

Speaker 2  1:46  
a name that's basically synonymous with optical illusion, impossible buildings and the kind of art that makes you question the very fabric of reality.

Speaker 1  1:54  
But I think for a lot of people, and I was in this camp for a while, Escher is just the guy who made those cool posters you see in like a college dorm room,

Unknown Speaker  2:03  
right? Or on the cover of a math textbook,

Unknown Speaker  2:05  
exactly. And our goal today,

Speaker 1  2:07  
our mission, is to show you that it is so, so much more than just trippy visuals. There's a profound depth there. There really is. That's the thing that really floored me when we started digging into the sources for this. You look at his work, these incredibly intricate patterns, these tessellations, these impossible geometries, mathematical singularities even, right? And you just assume, okay, this guy was a math prodigy. He must have been a genius. You think he was like calculating algorithms in his sleep to make these shapes fit so perfectly.

Speaker 2  2:40  
And that is the central paradox of escher's life. It's the big twist in the story, which is he had zero formal mathematical training, absolutely none. That just blows my mind. Unbelievable. If you look at his early life, he was actually, well, he was a bad student. He failed his exams. He did. He failed his high school exams. He wasn't a mathematician. He was an artist who just stumbled into these incredibly deep, complex mathematical concepts through what pure intuition, visual curiosity.

Speaker 1  3:08  
He said something amazing about that, didn't he something about a garden? Yeah, he

Speaker 2  3:11  
famously said he felt like he was wandering through a garden that mathematicians had built, but he'd found the gate completely by accident. Wow. He was just playing a game, he said. And it just so happened that the rules of his game were the rules of advanced geometry.

Speaker 1  3:26  
So that's the mission for today's deep touch. We want to figure out how a man who couldn't solve an equation on a chalkboard ended up becoming the most beloved, the most revered artist for mathematicians and scientists all over the world.

Speaker 2  3:39  
We're going to trace that journey, the journey from him drawing realistic landscapes, things he saw with his eyes exactly to what he called mental imagery, things that could only exist inside his head.

Speaker 1  3:51  
It's a move from observation to pure construction. Yeah. He stopped drawing the world and started building new ones.

Speaker 2  3:57  
He completely broke the boundaries between art and science. So get ready, this isn't

Speaker 1  4:01  
just about looking at pretty pictures. As Escher himself said, this is Brain gymnastics. We're going to be talking tessellations, impossible geometries and that deep psychological need. We all have to find some order in the chaos. Let's get into it.

Speaker 2  4:16  
Okay, so to really understand Escher, you have to go back to the beginning, back to his roots. Where did this all start? Well, he was born in the Netherlands in 1898 he was the youngest son, and his father was a civil engineer.

Speaker 1  4:30  
Oh, okay, see, that's interesting, right there? A civil engineer. So, structure, math, planning, it was in the family.

Speaker 2  4:38  
It was in the family. But like you said, the academic path was it was not smooth for him. He was a sickly kid, spent time in special schools, and he just did not click with the standard curriculum,

Speaker 1  4:49  
not at all. But I guess with his dad being an engineer, the family probably pushed him towards something

Speaker 2  4:53  
practical, right? They did. He actually started out studying architecture in Harlem, which

Speaker 1  4:57  
makes a certain kind of sense architecture. Is art, but it's art with rules. It's structural.

Speaker 2  5:03  
It is but that was the problem. It requires all these rigorous technical calculations. It demands a kind of mathematical brain that Escher just he didn't have it, and he didn't care for it, so he was struggling. Oh yeah, he was failing his classes. He was miserable. He was on the verge of dropping out.

Speaker 1  5:19  
But then something happened, a bit of fate, a teacher,

Speaker 2  5:24  
a graphic artist named Samuel jasrin de Mesquita.

Speaker 1  5:28  
And this teacher, de Mesquita, he saw something in Escher that the math teachers were completely missing.

Speaker 2  5:34  
He noticed that, okay, maybe Escher stinks at the engineering calculations, but his wood cuts, his Prince, his ability to see negative space and contrast was just extraordinary. He saw the raw talent. He saw a graphic artist, not an architect. He's the one who encouraged Escher to switch his major to the decorative arts, or what we would now call graphic arts,

Speaker 1  5:57  
and thank goodness he did. I mean, can you imagine a world where Escher just became some mediocre architect designing, you know, boring, functional office buildings in the

Speaker 2  6:06  
Netherlands, we would have missed out on all the space time bending. But do you

Speaker 1  6:10  
think that early training, that architectural thinking, do you think it stuck with him, because even his craziest, most impossible drawings still feel so solid they feel

Speaker 2  6:21  
built 100% he never lost that sense of structural integrity. His impossible buildings might break the laws of physics, but they obey the laws of drawing. That's a great way to put it. He's still using bricks and mortar and wood beams. He learned how to represent three dimensional space on a two dimensional plane in those classes, and that became the foundation for how he would eventually tear those rules apart.

Speaker 1  6:42  
So he switches to graphic arts. He finishes his studies, and then he does what a lot of young artists in that era did.

Unknown Speaker  6:48  
He went south. You went to Italy,

Speaker 2  6:50  
yeah, this is what's known as his Italian period. It's roughly from 1923 to 1935 he lived in Rome. He traveled all over the countryside. And if I were to show you a print he made in, say, 1928 you might not even recognize it as an Escher, right?

Speaker 1  7:05  
I was looking at some of these in our notes, specifically a piece called Castro Volvo from 1930 great example. It's a landscape, a beautiful, incredibly detailed lithograph of an Italian town built on a cliff. But it's realistic. There are no lizards turning into birds, no stairs going nowhere. It just looks like a really, really high end travel poster.

Speaker 2  7:26  
It is realistic, yes, but if you look closely, you can see the seeds of what's coming. You can see the obsessions starting

Speaker 1  7:31  
to form. How so, what should we be looking for? Look at the perspective in castrovalva.

Speaker 2  7:37  
He's playing with high and low angles at the same time. He's obsessed with the geometry of those Italian Hill towns, way the buildings stack on top of each other exactly the way a roof for one house is the foundation for the house above it the sharp contrasts of light and shadow. He wasn't interested in the feeling of the landscape like an impressionist would be. He was interested in the bones of it, the structure.

Speaker 1  8:00  
So he is observing the world, but he's looking for the Hidden Geometry inside the world. It feels less like art and more like

Speaker 2  8:10  
data collection. That is a perfect way to put it. He was building a library of forms in his head. He was capturing reality, not bending it, not yet. And he was happy there, right? He loved Italy, he did. But then the political climate changed, Mussolini and the rise of fascism. Escher had absolutely no interest in politics. He found the fanaticism deeply disturbing, and so he decided it was time

Speaker 1  8:30  
to leave. So he moves around for a bit, Switzerland, Belgium, and

Speaker 2  8:33  
eventually back to the Netherlands. And he hated it. He hated the weather, right? He missed the Italian Sun. He missed the dramatic landscapes. The Netherlands is flat. It's gray. So because he was deprived of that inspiration from the outside world, he was forced to turn inward. He had to start drawing from his imagination. But before that full turn inward, there was one critical moment, the the radioactive spider bite moment, if you will. That happened on a trip to Spain The Alhambra. The Alhambra. Now, Escher had first visited this Moorish palace in Granada, way back in 1922 as a young man, but he went back in 1936 and that second trip, that trip changed everything.

Speaker 1  9:13  
So for anyone listening who hasn't been there, set the scene for us. What is the vibe of the Alhambra? Why did this place just completely blow his mind.

Speaker 2  9:20  
Okay, so the Alhambra is this breathtaking masterpiece of Islamic architecture from the 13th and 14th centuries. But the thing that sets it apart, the thing Escher zeroed in on, is the tile work, the dots everywhere, the walls, the floors, the ceilings, they're covered in these infinite, repeating geometric patterns. Now in traditional Islamic art, you often avoid depicting living figures, humans, animals for religious reasons to avoid idolatry. So instead, the artisans became absolute masters of what's called the regular division of the plane.

Speaker 1  9:50  
Regular division of the plane. Okay, that sounds like a term straight out of a geometry textbook.

Speaker 2  9:55  
It is, but the concept is simple. It just means covering a surface with shapes so that there. No gaps and no overlaps, like tiling a bathroom floor, exactly like tiling a bathroom floor, but infinitely more complex. Instead of just boring squares, you have these intricate interlocking stars, hexagons, complex polygons that weave over and under each other.

Speaker 1  10:14  
And Escher walks in, he sees these walls just pulsating with these complex interlocking shapes, and he has his aha moment, a

Speaker 2  10:22  
total epiphany. He became obsessed. He and his wife spent days just sketching the tiles, copying the patterns. He called it the richest source of inspiration I have ever tapped. It wasn't just decoration to him. It was a puzzle, a profound mathematical puzzle, a revelation. It was a revelation. He realized that physical space could be filled in this incredibly disciplined mathematical way. But here is where Escher takes a sharp left turn away from that Islamic tradition. Okay. He wasn't content with just abstract shapes. He wanted to do something that the Moorish artist deliberately did not do. He wanted to use recognizable figures. He wanted his tiles to be lizards. He wanted them to be birds and fish and knights on horseback.

Unknown Speaker  11:05  
He took the mathematical rigor of

Speaker 1  11:07  
the regular division of the plane, but he turned it into

Unknown Speaker  11:11  
a game of metamorphosis

Speaker 2  11:12  
precisely, and that marks the big transition we talked about. He goes from observing the Italian landscape to creating mental imagery. He stops looking at the world outside his window and starts looking at the puzzle inside his own head. He wanted to see if he could make a lizard lock perfectly with another lizard with absolutely no empty space between them, which

Speaker 1  11:31  
leads us directly into, I mean, one of his most famous techniques, tessellation, right?

Speaker 2  11:37  
And a tessellation at its heart is just that tiling we mentioned, fitting shapes together perfectly, but Escher. Escher took it to this whole other psychological level. It wasn't just about the pattern, it was about the relationship between the shapes. We should

Speaker 1  11:50  
probably look at a specific example here, one of the heavy hitters. What about day and night?

Speaker 2  11:55  
Perfect choice from 1937 This was actually his most popular print ever. He sold more of this than any other. Okay, so walk us through it. What are we looking at? So it's this masterpiece of duality. You start at the bottom center of the image, and you see a checkerboard pattern of Dutch fields landscape.

Unknown Speaker  12:11  
Okay, pretty straightforward so far.

Speaker 2  12:13  
But as your eye moves up the image, those perfect squares begin to to warp. They transform, they stretch, they stretch, and they morph. The white daytime fields slowly become white birds, and they're flying off to the right into a black night sky. At the same time, the black fields morph into black birds flying to the left into a white daytime sky.

Speaker 1  12:37  
It's so clever because the birds are literally created out of the negative space of the other birds. Yeah, the shape of a white bird is defined by the black birds that surround it.

Speaker 2  12:48  
Yes, and that's a classic psychology problem, right? The figure ground problem explain that a little our brains are wired to see a foreground object, the figure and a background. We can't usually see both at the same time. You focus on a person's face, not the wall behind them, but in day and night, Escher forces your brain to switch back and forth, back and forth. You see the white birds, then you blink, and suddenly you see the black birds. It creates this visual vibration,

Speaker 1  13:13  
and the theme is so rich. It's not just a cool pattern. It's day and night. It's earth and sky, black and white, left and right. It's like he's saying that these opposites aren't really separate things at all.

Speaker 2  13:23  
They're two sides of the same coin, part of the same fabric. You can't have one without the other, the merging of opposites. And he pushes this idea even further a year later, in 1938 with a piece called sky and water time.

Unknown Speaker  13:34  
Oh, I love that one.

Speaker 1  13:36  
That's the one where at the very top it's clearly black birds flying in white air, yep, and at the very bottom, it's clearly white fish swimming in black water.

Speaker 2  13:45  
But what about the middle? The middle is the magic part. That's the interface.

Speaker 1  13:50  
In the middle, the birds and the fish are kind of the same thing. The air becomes the water, the bird becomes the fish.

Speaker 2  13:55  
It's biologically impossible. Of course, birds do not turn into fish, right? But in the visual language that Escher has created, it feels inevitable. As your eye scans down the print, the background slowly solidifies and becomes the foreground.

Speaker 1  14:13  
It's like he's pointing out that the only real difference between a bird and the air it's flying in is where you choose to focus your attention. It feels almost philosophical, like something out of Zen Buddhism.

Unknown Speaker  14:23  
You know, everything is connected.

Speaker 2  14:25  
It is. It's all about challenging the boundaries we put between things. And he takes this concept of transformation to its absolute logical extreme in his metamorphosis series,

Speaker 1  14:35  
oh man, especially metamorphosis to second that thing is, it's epic. It's massive.

Speaker 2  14:40  
It's a four meter long woodcut. It's a loop. If you were to unroll it, it would span an entire wall.

Speaker 1  14:46  
It's basically a comic strip, a narrative it is. It starts

Speaker 2  14:49  
with the word metamorphose the letters of the word turn into black and white squares. Those squares turn into a checkerboard, and the checkerboard becomes lizards. The lizards become a. Honeycomb. The honeycomb becomes bees. The bees turn into birds. The birds then transform into a cityscape of a Trani, that little town in Italy he loves so much.

Speaker 1  15:11  
And then the city turns back into blocks. The blocks turn back into the checkerboard, and the checkerboard turns back into the word metamorphosis.

Speaker 2  15:18  
It's a complete cycle. The beginning is the end. The end is the beginning, a strange loop, a perfect, Strange Loop. And this is the point where the mathematicians really started to sit up and pay attention,

Unknown Speaker  15:28  
because he was doing their work without knowing it exactly.

Speaker 2  15:31  
He was intuitively using these complex geometric principles, things like translations, rotations, reflections, glide reflections, to make all these shapes fit together perfectly, but he didn't know the formal terms. Had no idea he was just, as he called it, playing. He was just trying to solve the visual puzzle, but in doing so, he was solving complex geometric problems that mathematicians study for years in university.

Speaker 1  15:55  
I love that image, just him in his studio, carving these wood blocks, solving high level geometry just because he wanted the lizard's nose to fit perfectly into the crook of the other lizard's knee.

Speaker 2  16:06  
He wasn't trying to prove a theorem or write a paper. He's just trying to satisfy this visual itch he had. It really speaks to the unintentional genius part of his story. He wrote to a friend once that he felt these repeating shapes were almost independent of him, like they had a will of their own, and he was just the one uncovering them.

Speaker 1  16:24  
Wow. So we've got tessellations, this division of a flat, two dimensional plane, but Escher wasn't satisfied with just two dimensions. He wanted to bring that same sense of order to 3d space,

Speaker 2  16:35  
which brings us to his obsession with crystals and polyhedra, right?

Speaker 1  16:39  
This is section three in our outline, and it's so important for understanding

Unknown Speaker  16:43  
his personality. The man craved order.

Speaker 2  16:45  
And think about the time he was living in two world wars, economic collapse. The world around him was pure chaos, but in mathematics, and specifically in these things called platonic solids, he found something that was certain something undeniable.

Speaker 1  17:02  
Okay, quick refresher, for those of us who might have snoozed through geometry class, what are the platonic solids? They're the

Speaker 2  17:08  
perfect 3d shapes, and there are only five of them in the entire universe. You've got the tetrahedron, which is a four sided pyramid, okay, the cube, obviously the octahedron, which has eight sides, the dodecahedron with 12 sides, and the icosahedron with 20

Unknown Speaker  17:22  
and what makes them so special, so perfect? What makes

Speaker 2  17:25  
them special is that every single face is the exact same regular polygon, and every single corner is identical. They represent perfect symmetry. Escher called them rock hard reality.

Speaker 1  17:35  
Rock hard reality. I love that phrase in a world where everything else is falling apart, politics, society, war, a cube is always

Unknown Speaker  17:43  
a cube. It's a source of stability. It's

Speaker 2  17:45  
a refuge from the chaos. And you can see this obsession so clearly in a piece called stars from 1948

Speaker 1  17:51  
Ah, yes, the one that looks like it's from a sci fi movie. It's a wood engraving.

Speaker 2  17:55  
And you see all these different geometric solids floating in deep space, like strange planets or asteroids. It looks very cool, very mathematical. But then there's the classic Escher twist, the chameleons. The chameleons right inside the central shape, which is a complex structure of three interlocking octahedra. He places these little chameleons just crawling around.

Speaker 1  18:17  
So why chameleons? What's the symbolism there? Is it? Because they change color.

Speaker 2  18:20  
That's definitely one interpretation. The chameleons adapt. They are mutable. They represent life. The crystal is immutable. It represents logic. So you have this stark contrast, the

Speaker 1  18:31  
cold, perfect, unchanging, mathematical form versus the living, adaptable, maybe even chaotic creature that's trapped inside it.

Speaker 2  18:39  
And it raises a question, right? Is the chameleon being protected by the mathematical structure, or is it imprisoned by it?

Speaker 1  18:45  
That's a great question, and it connects directly to another piece ordering chaos from 1950 the title of that one is, well, it's a little less subtle, a bit

Speaker 2  18:52  
more on the nose, yeah, but it's a beautiful lithograph. In the center, you have a stellated dodecahedron, basically a star shaped crystal, and it's resting inside a perfect, clear glass sphere. It's pristine, flawless.

Speaker 1  19:09  
And surrounding this perfect sphere of order is junk,

Speaker 2  19:12  
total junk. It's trash. It looks like broken bits of pipe, a crumpled sardine can, random debris from daily life.

Speaker 1  19:19  
The symbolism there is pretty potent. The crystal is the eternal, unchangeable truth of mathematics, and all the junk around it is just the messy, temporary garbage of our daily lives,

Speaker 2  19:30  
and the crystal is safe inside its sphere. It's untouched by all the mess.

Speaker 1  19:33  
It's almost comforting. It's like he's saying, no matter how messy my life gets, no matter how chaotic the world is, this geometric truth still exists. It's like a religious conviction for him, but his god is geometry.

Speaker 2  19:44  
But here's the thing about Escher, he wasn't content to just admire the order. He had to break it. He wanted to see what happens when you take the rules of 3d space and just twist them until they snap.

Unknown Speaker  19:54  
And this is where we get into the impossible worlds. This

Speaker 2  19:57  
is the stuff that made him a household name, the crazy stairs. The crazy stairs. We have to talk about relativity from 1953

Speaker 1  20:04  
okay, let's paint the picture for everyone. You're looking at this room, but it's not a normal room. It doesn't have a clear floor or ceiling. It's like there are three different sources of gravity all working at the same time.

Speaker 2  20:16  
That's it, exactly. You have these faceless, generic figures just walking around. And for one figure, a particular surface is their floor, but for another figure, that exact same surface is a wall they're walking past. And for a third It might even be a ceiling, and

Speaker 1  20:34  
they're all in the same image, but they can never meet. They can't interact. I mean, this image is everywhere. I remember from the movie labyrinth with David Bowie Inception used it. Squid game had a huge set piece based on it. It's just embedded in our culture.

Unknown Speaker  20:47  
Why do you think this one image sticks with us so much?

Speaker 2  20:50  
I think it's because it fundamentally challenges how our brains process perspective. We are hardwired for one source of gravity down is down. But here, Escher gives us three different downs, and each individual part makes sense. That's the key. You look at one staircase, and it's fine. You look at another, it's fine. But when your brain tries to combine them into a single coherent whole, it just glitches. It can't resolve the paradox.

Speaker 1  21:17  
He was making fun of gravity, as he put it, it feels, I don't know, a little lonely though, all these people walking right past each other, but living in totally different dimensions.

Speaker 2  21:26  
That's a really interesting reading of it. And this whole line of thinking, these impossible structures, it leads us to a really important feedback loop in his life, the Penrose connection. Okay, we mentioned earlier that Escher influenced mathematicians. Well, this is a case where the influence

Speaker 1  21:42  
went both ways, right? This is Roger Penrose, the famous mathematician and physicist. He won a Nobel Prize.

Speaker 2  21:47  
That's the one. So Roger Penrose saw escher's work at an exhibition in Amsterdam, and was just completely blown away by it. He went home and started trying to draw his own impossible objects, just for fun, and he created something famous, right? He created what is now known as the Penrose triangle. It's a drawing of a triangle that looks solid, but it can't possibly exist in 3d space. Each corner is a 90 degree angle, but they all connect to form a loop, which is impossible. And Penrose sent a copy of his drawing to Escher,

Speaker 1  22:18  
and Escher sees this simple line drawing, and basically says, hold my beer.

Speaker 2  22:22  
Pretty much. Escher took that core concept of the Penrose triangle, which was just an abstract shape, and he built an entire world, an entire narrative around it, and that's how we got the print ascending and descending in 1960

Speaker 1  22:35  
Ah, yes, the one with the monks on the roof of a building, walking

Speaker 2  22:39  
on a never ending staircase. The monks are marching in the solemn procession in a loop. If you follow any one monk, their feet are constantly stepping up, up, up, but they never get any higher. They end up right back where they started. It's the Stairmaster from hell. It's such a powerful metaphor for futility. And Escher actually wrote about this piece. He talked about the two recalcitrant individuals who refused to participate. If you look closely at the print, there are two figures who are not on the stairs.

Speaker 1  23:05  
Oh, right. One is just sitting on a terrace looking off into space, and another's on a lower balcony.

Speaker 2  23:11  
So everyone else is marching in this endless futile loop of progress that goes absolutely nowhere, and these two guys are just sitting it out, watching

Speaker 1  23:20  
it makes you ask, is the climb meaningful, if it never actually gets you any higher? Is it a commentary on the absurdity of human striving?

Speaker 2  23:27  
He was trying to explore the logic of space, but he ended up stumbling into deep philosophy,

Speaker 1  23:32  
and he did it again a year later with waterfall in 1961 this is another one of his famous perpetual motion machines.

Speaker 2  23:40  
Yes, in this one, you see water flowing down a channel. It falls over a water wheel, causing it to turn, and then somehow the water flows along a flat aqueduct that leads it right back to the top of the waterfall to start the journey all over

Unknown Speaker  23:53  
again. It's using two Penrose triangles to trick your eye, right? Exactly?

Speaker 2  23:57  
It's a visual impossibility, but it looks so plausible on the page.

Speaker 1  24:01  
I think that's the key to his genius. It looks plausible. He draws it with such incredible realistic detail, the brickwork on the towers, the little plants growing in the garden, the woman hanging laundry in the background, that you trust Him, you believe the world he's showing you. And then he

Speaker 2  24:18  
betrays that trust with the impossible geometry. He surrounds the mystery with the mundane. That's just true. If the whole environment looked alien and weird, you'd just dismiss it as fantasy. But because it looks like a normal Italian building, the impossibility hits you that much harder.

Speaker 1  24:32  
Okay, so we've covered tessellations on a 2d plane. We've covered perfect crystals and impossible buildings in 3d Yeah, but there was one final frontier that Escher was desperate to conquer,

Speaker 2  24:44  
the infinite infinity. And infinity is a real headache for a visual artist.

Speaker 1  24:48  
Why? Because a piece of paper has edges. It's finite. How do you possibly show something that goes on forever on a sheet of paper that is, you know, 12 inches wide, right?

Speaker 2  24:58  
You can't just draw a really, really. Long line and let it run off the page. That's cheating. It's cheating. And Escher struggled with this for years. He tried different techniques, like fading things out at the edges or blurring them, but he was never satisfied. It felt like a cop out. He wanted a rigorous mathematical solution to the

Speaker 1  25:16  
problem, and once again, a mathematician came to the rescue, this time, a guy named HSM Coxeter.

Speaker 2  25:22  
Coxeter another giant in the world of geometry. So what was Coxeter solution? Coxeter sent Escher a diagram of something called a hyperbolic tessellation. Now, without getting too deep into the weeds of the math, please, yeah. Keep it simple for us. Imagine a circle. This is called the Poincare disk model of hyperbolic space. Yep, in this model, the very center of the circle is normal, but as you move out from the center towards the edge, the space itself begins to compress, like it's shrinking exactly in hyperbolic geometry, the space gets denser and denser the closer you get to the boundary. So if you were a little creature living inside that circle, and you started to walk toward the edge, you would start to shrink. Your steps would get smaller and smaller. You could walk forever and ever, and you would never, ever actually reach the edge.

Speaker 1  26:09  
So from your perspective inside the circle, your universe is infinite,

Speaker 2  26:14  
but from our perspective, looking at the drawing, it's all contained neatly inside a finite circle.

Speaker 1  26:19  
That is absolutely genius. It's a way to trap infinity inside a boundary.

Speaker 2  26:23  
And Escher saw this diagram, and his mind just exploded. He finally had his solution, and he used it to create the circle limit series, specifically circle limit three, yes, from 1959 it's a woodcut that depicts fish. There are chains of fish swimming from the center of the circle outwards. As they get closer to the edge, they get infinitely smaller. They fit together

Speaker 1  26:46  
perfectly, just like his other tessellations, but they diminish.

Speaker 2  26:50  
It's a finite image that contains an infinite universe of fish.

Speaker 1  26:54  
Just looking at it gives you a sense of vertigo. It feels like you're falling into the center or being pulled out to the edge.

Speaker 2  26:59  
It does, and he kept working on this concept right up until the very end of his life. His final print, which he created in 1969 just a few years before he died, is called Snakes. Snakes. Another woodcut right? An incredibly complex one. It features these snakes winding their way through a series of interlocking rings, and the rings shrink endlessly as they approach the center and as they approach the edge. So it's double infinity. It's infinity in both directions. And think about this. At the end of his life, his health was failing, his hands were shaking, and he was still meticulously carving this impossible vision of infinity. He was obsessed with finding the language of matter, space and the universe.

Speaker 1  27:38  
That level of dedication is just incredible. He was always chasing perfection, and speaking of perfection, that brings up another mathematical concept that's often tied to art and nature, golden ratio and the Fibonacci sequence.

Speaker 2  27:50  
This is a great connection, because it brings Escher back from the abstract world of pure math to the natural world. We've talked so much about his mental imagery, but he never stopped being a keen observer of nature.

Speaker 1  28:02  
So quick refresher again for our listeners, the Fibonacci sequence is that famous series of numbers, 0112358, and so on. Each number is just the sum of the two before it.

Speaker 2  28:12  
And if you visualize that sequence geometrically, if you make squares with those side links, you get a beautiful spiral, the golden spiral, and you see this pattern everywhere

Unknown Speaker  28:22  
in nature, pine cones, sunflowers, Nautilus shells.

Speaker 2  28:26  
It represents a kind of esthetic perfection, a growth pattern that is both efficient and

Speaker 1  28:31  
beautiful. And you think Escher was consciously using this.

Speaker 2  28:34  
I think he was absolutely aware of it, those spirals in his final print, snakes, that's recursive symmetry. He was tapping into the exact same mathematical principles that a sunflower uses to pack its seeds as tightly as possible. He was fascinated by this idea of recursive symmetry, patterns within patterns.

Speaker 1  28:52  
It's kind of related to chaos theory, isn't it, the idea of a strange attractor, in a way.

Speaker 2  28:57  
Yes, it's about finding the hidden order inside what appears to be randomness. Escher's work is often seen now as a precursor to fractal art, right?

Speaker 1  29:05  
Those computer generated images like the Mandelbrot set, where you can zoom in forever and the same basic shape just keeps repeating exactly.

Speaker 2  29:13  
But Escher didn't have a supercomputer. He didn't have a GPU to render these complex fractals. He did it all by hand with wood and a carving gouge, but the intuition behind it was the same. He was visualizing recursion long before we had the machines to simulate it properly. That's what blows my mind. He was a human computer, a human computer with a soul, and that soulfulness, that introspection, really comes out when he starts dealing with the concept of the self. Which brings us to our next section,

Speaker 1  29:41  
reflections, mirrors. The man loved mirrors and reflections.

Speaker 2  29:45  
He did. There's a very famous lithograph from 1935 called hand with reflecting sphere. You've definitely seen it. Oh yeah, it's

Unknown Speaker  29:52  
him holding a perfectly mirrored silver ball, right?

Speaker 2  29:55  
And in the reflection of the ball, you can see his own face. You can see the room he's in, the furniture, the scene. Healing beams.

Speaker 1  30:00  
But think about the perspective for a second, the self eshers face is at the absolute dead center of this reflected universe. But then look at the real world outside the sphere. What about it? The hand that's holding the sphere. It isn't attached to a body. It's just a hand floating in a white void. The only reality in the image is what's inside the reflection.

Speaker 2  30:22  
So the reflection has more detail, more substance, than the reality that is supposedly holding it. That's a complete reversal.

Speaker 1  30:29  
It's a way of questioning what's real is the world inside the sphere the true reality. It creates this perfect closed loop of self reference. But if

Speaker 2  30:38  
we're going to talk about loops and self reference, we have to talk about the ultimate brain melter. Oh boy. Print gallery. Print Gallery from 1956 this is, in my opinion, his most complex intellectual achievement. This is the final boss battle of Escher Prince.

Speaker 1  30:53  
Okay, describe this one for us, and let's go slow, because this one is tricky to wrap your head around.

Speaker 2  30:58  
Okay, so on the surface, you see young man. He's standing in an art gallery. Simple enough, he's looking at a picture on the wall. The picture is of a harbor town, maybe in Malta. You can see a ship in the harbor. You can see the buildings along the dock.

Speaker 1  31:11  
Okay, still falling guy in a gallery looking at a boat. Picture normal,

Speaker 2  31:14  
but now follow the buildings in the picture. As your eye moves to the right, the buildings start to curve, and they get larger. They start to wrap around the frame of the image. They're distorting. They're distorting and expanding until the buildings in the picture become the very walls of the gallery in which the young man is standing Wait. Say that again, he is standing in a gallery looking at a picture of the gallery that he is currently standing in.

Speaker 1  31:41  
So he's looking at himself, but from the outside. Yes, the picture contains the viewer.

Speaker 2  31:46  
Yes, it's what's known as the Drost effect, a picture within a picture, but Asher has twisted it into an impossible spiral. The outside becomes the inside, which becomes the outside.

Speaker 1  31:58  
My brain hurts, but what happens in the very center of the spiral? If it's twisting in on itself, surely it gets all messed up in the middle.

Speaker 2  32:06  
That is the singularity in the original print. There's a blurry white spot right in the middle, yeah. Escher actually put his signature there to cover it up because he knew the math broke down at that point. He couldn't draw the center because logically, the center would have to contain the entire image all over again, but infinitely smaller. It was a paradox he couldn't resolve with a pencil.

Speaker 1  32:26  
And mathematicians actually went back and analyzed this later, didn't they?

Speaker 2  32:29  
They did decades later, a team of mathematicians at Leiden University in the Netherlands figured out the complex grid structure Escher must have used. They found that it contains a true mathematical singularity, a hole in the fabric of the space he created. And they used a computer to fill it in. They used a computer to fix the hole that Escher had to leave blank. They completed the spiral. It turns out that Escher had intuitively constructed a visual representation of a Riemann surface, a concept from very advanced mathematics.

Speaker 1  33:01  
Again, the guy who failed high school math, he essentially drew a visual wormhole by hand.

Speaker 2  33:07  
It just proves that mathematics isn't only about calculation, it's about imagination. He could visualize these concepts that usually require pages and pages of complex calculus to even describe.

Speaker 1  33:18  
And this connects directly to that famous book Douglas hofstadter's Good old Escher Bach,

Speaker 2  33:23  
a seminal Pulitzer Prize winning book, Hofstadter uses escher's work as a visual guide to explain these incredibly abstract ideas like strange loops and the nature of consciousness. What's the connection to consciousness? The idea is that consciousness itself is a self referential loop. The feeling of I comes from a system that can look back on itself. I am thinking about myself, thinking

Speaker 1  33:46  
so drawing hands, the hands that draw each other into existence, is the perfect visual metaphor for self awareness. We create ourselves.

Speaker 2  33:55  
And print gallery is the same idea. The observer is fundamentally part of the system being observed. You can't separate them. That is,

Speaker 1  34:02  
that is incredibly deep. It stops being about cool drawings and starts being about the fundamental nature of how we exist.

Speaker 2  34:08  
It's philosophy disguised as art. So we have this absolute

Speaker 1  34:11  
genius, this artist, mathematician, philosopher. What was he like as a person? You'd expect him to be this wild, trippy character, right? A wizard living in a tower

Speaker 2  34:19  
somewhere, and ironically, not at all. That brings us to our last section, the man behind the mind benders was, well, he was a bit of a square, really, a square. Moritz Escher was meticulous. He was obsessive. He dressed neatly. He was a dutiful husband and father. He didn't consider himself a visionary artist in that grand, romantic sense. He said he was just wandering around in enigmas. He was very humble, almost to a fault.

Speaker 1  34:47  
It's that famous story about Mick Jagger. I feel like that sums up his personality, purpose. Oh, no,

Speaker 2  34:52  
this story is a classic. Okay, so it's the late 1960s the Rolling Stones are the biggest band in the world. The counterculture, the hippies, they act. Absolutely love escher's work. They think it's the coolest, most psychedelic thing ever, right? So Mick Jagger writes a letter to MC Azure, asking him to design an album cover for the stones. I think it was for through the past darkly

Speaker 1  35:12  
and Esher designed Rolling Stones cover that would have been one of the most iconic album covers of all time. It belongs in a museum.

Speaker 2  35:18  
You would think, yeah. But Mick Jagger makes one crucial mistake. He starts the letter with dear Moritz.

Speaker 1  35:24  
Uh oh, too informal, like they're old buddies.

Speaker 2  35:27  
Way too informal for Escher. He wrote back a very curt letter refusing the commission, and he told Jagger's assistant to and I quote, please tell Mr. Jagger, I am not Moritz.

Speaker 1  35:37  
To him, ouch. I am not Moritz. Diem. That is ice cold, but

Speaker 2  35:41  
it shows his complete detachment from pop culture and fame. The hippies adored him. They saw his work as these incredible psychedelic visions. They assumed he must be tripping on acid to come up with these things, but he wasn't, not at all. Esher didn't do drugs. He barely drank. He honestly didn't understand why all these long haired kids in America liked his work so much. He was a man obsessed with order and logic. They just saw the trippy visuals. It was a total mismatch.

Speaker 1  36:09  
It's like he was an orderly man creating chaos, and they were chaotic people who were

Unknown Speaker  36:13  
consuming his order.

Speaker 2  36:14  
That's a fascinating way to look at it. But his real legacy goes way beyond album covers. He didn't design. It's in science classrooms. It really is. He's a staple crystallography, topology, hyperbolic space science textbooks. Uses art all the time, because his pictures explain these incredibly complex concepts better than any paragraph of words ever could. If you want to explain symmetry to a chemistry student, you don't give them an equation, you show them an Escher print.

Speaker 1  36:43  
And in the art world, he heavily influenced the optical art movement of the 60s. Even though he hated the

Speaker 2  36:50  
label, he rejected all labels. He once said, I don't belong anywhere. And in a way, he was right. He wasn't really an artist, he wasn't really a mathematician. He created his own category.

Speaker 1  36:59  
So as we start to wrap up this deep dive. Let's try to synthesize all of this. We've gone from the tiles of the Alhambra to the infinite edge of a circle. We've seen birds turn into fish and water flow uphill. What's the big takeaway here?

Speaker 2  37:10  
I think Escher showed us that reality is well, it's softer than we think it is. He proved visually that our perception of the world is fragile and easily tricked. He said, order is repetition of units and chaos is multiplicity without rhythm.

Speaker 1  37:25  
And he spent his entire life trying to find the rhythm and the multiplicity Exactly.

Speaker 2  37:30  
He taught us that the hard boundaries we see everywhere between foreground and background, up and down, inside and outside, are often just illusions created by our own minds. I really

Speaker 1  37:40  
like that. Finding the rhythm, it makes the chaos of the world seem more manageable, if you can just find the underlying pattern.

Speaker 2  37:46  
And I think for you, for the listener, the so what of all this is that Escher encourages us, he challenges us to look at the world from multiple perspectives at the same time

Speaker 1  37:58  
in relativity, gravity depends on where you're standing in day and night. Whether you see a bird or not depends on whether you focus on

Unknown Speaker  38:05  
the light or the dark.

Speaker 2  38:06  
It's an exercise in cognitive flexibility. It's brain gymnastics.

Speaker 1  38:10  
It really teaches us to question our own assumptions. If a floor can become a ceiling, then maybe the things we think of as rock hard reality in our own lives are actually just a matter of perspective.

Speaker 2  38:21  
Maybe that problem you're stuck on is just a pattern you haven't recognized yet. He gives

Speaker 1  38:25  
us the tools to see those hidden connections. He does. I want to leave everyone with a final thought to chew on, something to mull over. We talked about metamorphosis, where rigid squares can transform into living birds. So look at your own life. What are the rigid squares that are just waiting to turn into birds? What's the chaos you're dealing with? That might actually be a pattern you just haven't figured

Speaker 2  38:48  
out yet, because, as Escher himself said, Only those who attempt the absurd will achieve the impossible.

Speaker 1  38:54  
Thanks for diving dump with us. Keep looking for the pattern. See you next time.

Transcribed by https://otter.ai