The Joy of Why - What Can Tiling Patterns Teach Us?

Episode Date: July 3, 2024

In the tiling of wallpaper and bathroom floors, collective repeated patterns often emerge. Mathematicians have long tried to find a tiling shape that never repeats in this way. In 2023, they lauded an unexpected amateur victor. That discovery of the elusive aperiodic monotile propelled the field into new dimensions. The study of tessellation is much more than a fun thought exercise: Peculiar, rare tiling formations can sometimes seem to tell us something about the natural world, from the structure of minerals to the organization of the cosmos. In this episode, co-host Janna Levin speaks with mathematician Natalie Priebe Frank on the subject of these complex geometric combinations, and where they may pop up unexpectedly. Specifically, they explore her research into quasicrystals — crystals that, like aperiodic tiles, enigmatically resist structural uniformity.

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What is better, cats or dogs? Listen wherever you get your podcasts. For thousands of years, cultures around the world have explored patterns with tiles, from ancient mosaics to stone floors to modern subway tiles. Perhaps for just as long, mathematicians have been enthralled by the way different shapes can or can't fit together to cover a plane, like a tabletop or a wall, and by the complex geometric combinations they've uncovered. For decades, the search was on for a single tile that might fill a plane in an aperiodic pattern, which is to say, one that has no repeats. pattern, which is to say one that has no repeats. Then in March 2023, David Smith, an amateur tiling enthusiast, found a single tile that resisted repetition, at least as far as his attempts could take him. Could this be the elusive, aperiodic monotile?

I'm Janna Levin, and this is The Joy of Why, a podcast from Quantum Magazine, where I take turns at the mic with my co-host, Steve Strogatz, exploring the biggest questions in math and science today. In today's episode, we'll speak with Natalie Prriby-Frank to ask why the discovery of the aperiodic monotile is so significant and what tessellations, which are tight arrangements of shapes over surfaces, what tessellations might reveal about the natural world. Natalie is a professor of mathematics and statistics at Vassar College. Her research is primarily on mathematical models for physical solids, and her research is also on

quasicrystals, which are ordered materials that resemble crystals but lack a consistently repeating structure. Natalie, we're so glad to have you with us and that you're bringing your expertise to this conversation. Thank you. It's great to be here. I wanted to begin with something very familiar. Anyone who has studied a bathroom floor or maybe wall tiles might have noticed that if they're made out of a single tile, that if they're made out of a single tile, they can only really be four-sided generalized rectangles or six-sided hexagons. And why do these shapes work for tiling a flat surface while others don't? Well, if you are making a periodic tiling, which is like a tiling with squares, sort of like an infinite checkerboard.

There's only certain ways you can do it as you observed. You could do squares, you can do hexagons, and there's one more you can do it with triangles. And that's because of what's known as the crystallographic restriction, which says that periodic tilings and structures can have rotational symmetries that are either two-fold, three-fold, four-fold, or six-fold. And that's it. Something with five-fold rotational symmetry cannot be achieved using a periodic tiling of the plane. So for people who don't know what a twofold or threefold symmetry is, you start with a particular shape and you're imagining rotating it. Is that right? Yeah. So if I were to take a square, then a square has fourfold rotational symmetry. You can kind of imagine spinning it around four times and that gets you back to where

it started. An equilateral triangle is going to have threefold rotational symmetry because you can have it land perfectly on top of itself three times and get yourself back to the starting point. So there's a relationship between the symmetry of the single tile that you're trying to use to cover the plane. Now, what if I tried to use an eight-sided octagon? If I tried to piece together the octagons to tile my bathroom floor, I'm halfway through this project, I'm very frustrated. What have I done wrong? Right. In the case of an octagon, it's that the angles are too big. If you take two octagons and connect them along an edge, then you're going to find a little wedge shape in between that you won't be able to fit a third octagon inside of. It's perfect for putting a square in, however,

so that's why you see lovely octagon square patterns all over the place. For periodic tilings, even if you use more than one tile shape, you're still not going to get eightfold symmetry. So even when I'm all done making my tiling out of octagons and squares, if I've made it as a periodic tiling, a repeating tiling, as a periodic tiling, a repeating tiling, then it won't have eightfold rotational symmetry. When you try to rotate by an eighth of a circle, the tiling will not line up with itself properly. Now, as much as people have obsessed about these perfect patterns, these periodic repeating patterns, mathematicians have also obsessed about breaking these patterns, these periodic repeating patterns. Mathematicians were also obsessed about breaking these patterns through aperiodic tilings, that is, patterns that in fact never repeat. Can you describe to us why it was so surprising that a truly aperiodic tiling might exist? Yeah, so this is a really foundational story for the field. It comes from the

philosophy of logic, actually, symbolic logic from back in the 1960s. Computers were coming to the fore, and people were asking questions about what the limits of computers and computation were going to be. And in particular, a logician named Hao Wang wrote this wonderful paper, and it appears in Scientific American, and he poses the following question. colored the edges so that when you use them to make a tiling, you're only allowed to put together squares along edges where the colors match. And suppose someone gives me a set of five of these square tiles, which are known as Wang tiles. And I have infinitely many copies of every type of the tiles, but there's only five basic types. Can I determine whether those particular five Wang tiles can actually form a tiling of the plane, or can I not? Seems like a simple problem, but the hard part is devising a single algorithm that could decide it for any given input of Wang

tiles. This is a decidability question. It's a very important sort of computational type of question. And Wang worked on this question and determined that the answer would be yes, this tiling problem would be decidable by an algorithm if and only if every set of Wang tiles that was capable of making a tiling of the plane was capable of making a periodic tiling of the plane. So this computational decidability question of can I write an algorithm to determine if any set of Wang tiles can or cannot tile became the question, can every finite set of Wang tiles that can make a tiling always make a periodic tiling? And he was skeptical that the answer to that could be yes.

Within a few years, one of Wang's students, Robert Berger, found an answer to the domino problem, as it's sometimes called. He found an answer by finding a set of aperiodic tiles. Berger's set, though, was made up of 20,426 tiles. It was a huge number of tiles. It is a lot. People immediately thought, of course, can we do better? Can we get smaller aperiodic sets of tiles? And that's when the, well when the race was on. And very quickly, another logician got it down to six Robinson tiles are basically square tiles. They have little bumps sticking out of some of the edges and some of the corners are cut off. Essentially, the tiles, when you stick them together, the geometry on this local level forces them to create a sort of a plus sign shaped pattern.

And then those can only be assembled into larger plus sign shaped patterns. You can't shift it over and have it land perfectly on top of itself again. And so you can geometrically see that these tiles can make a tiling, but that when you're done, the tiling will be forced to be non-periodic. Roger Penrose, he found out about this race to find smaller and smaller sets of aperiodic tiles, and he had a go at it. And he came up with what's now the famous Penrose tilings, which is an aperiodic set of two tiles. And one of the famous versions, the tiles are both rhombuses and one is thick and one is thin. And those are the ones that tend to show up in architecture and science centers all across the planet.

And Escher drawings. Yeah, and Escher-like drawings. And so when Penrose made this discovery, the tiles that he used, the angles are basically one-tenth of the circle. circle. So there's tenfold or maybe fivefold rotational symmetry, which notably is not possible for periodic tilings, as we mentioned in the crystallographic restrictions. So the tilings weren't going to be periodic anyway because they're aperiodic, but a crystallographer at the time, this is now in the 70s, looked at this Penrose tiling and said, boy, that is an awfully organized, non-periodic object. That thing looks almost like a crystal to me. I wonder what its diffraction pattern would look like.

And so what he did was he had someone do one of those lightbox diffraction experiments where you poke holes in the paper. And so he had them go on a Penrose tiling and poke holes in all the vertices in a large patch of tiles and did the lightbox experiment. And what emerged was a wonderful display of very bright dots in a five-holed rotational symmetry. And when, a few years later, actual physical quasicrystals were discovered in the lab in 1982, they discovered that the diffraction pattern looked nearly identical to that of the Penrose tiling. So this was a fun moment where a mathematical discovery foreshadowed an upcoming discovery in the physical sciences.

And that discovery was made by Daniel Schechtman, who got the Nobel Prize eventually in chemistry for the discovery of quasicrystals. in chemistry for the discovery of quasicrystals. Over the course of a few decades, we went from Wang's domino problem to Berger's huge aperiodic tile set, then to Penrose's dramatically reduced two-tile set. And it's all essentially been purely mathematical fun and games. That is, until Schechtman, a material scientist, comes along and shows that atoms within quasicrystals are essentially arranged in a Penrose pattern. It's a big moment, and it sort of reignites the race to find a single, aperiodic monotile. This hypothesized tile, the Einstein tile as it came to be known, a pun on the German term for one stone, might tell us something about the natural world. Around 2010, another amateur mathematician, Joan Taylor, she found a tile. It is an aperiodic monotile, but it has a flaw that some found disqualifying, which is that it's not connected.

It's sort of a central tile and then several satellite pieces that still are part of the tile. And so she did accomplish constructing a set in the Euclidean plain that can tile and only non-periodically. But because it wasn't connected, it didn't really do it for everyone. And she was just a hobbyist. Yeah, I mean, a lot of people really like tiling problems. They're fun. And there's online communities of people who are really connected to one another. And there are strategies that you can go about solving this kind of problem. Joan Taylor, she knew all about Penrose's work. She had studied him extensively. It's my understanding that she had actually gone for a bit of graduate school in mathematics. So she was very serious about it.

But she still, at the end of the day, was out there on her llama farm solving the Einstein problem. As you do. As you do. So that was really neat. And we were very excited when that happened. We'll be right back after this message. Welcome back to The Joy of Why. Math is such a challenging subject. It's often considered to be the subject of exceptional minds, but it also has this citizen science aspect to it that is really

fascinating. Before the break, we talked about Joan Taylor, an amateur who created a kind of aperiodic monotile that wasn't exactly what people were looking for. Then another amateur, David Smith, as we mentioned in the beginning of the episode, had his own monotile breakthrough in 2023, and it seemed much closer to the mark. As a print technician in England, he did this without all the infrastructure of an academic career. I asked our guest, Natalie Preeby-Frank, about his process. was looking for interesting tiles in the following way that's commonly practiced. So what you do, you can start with a hexagonal tiling of the plane. So it's just all hexagons. And then put a dot in the center of the hexagon and draw a line from that dot in the center to the middle of each edge, not the vertices, but the middle of each edge. And so you end up making the hexagon looks like six little kite-shaped things stuck together.

They're called polykites. And so what you do is you take those little kite-shaped things and you just simply regroup them. And you just say, okay, now that set of eight of them that are all stuck together, that's my tile. Let me see if I can tile with it. And so people have categorically gone through the low levels of all the possible combinations of those things. Someone in the literature a few years prior had pointed out a tile with a very large corona or possibly a tile that didn't tile. And so David

started playing with this particular version. Yes. And he called his the hat because it vaguely sort of looked like a big top hat. They decided it looked like a hat. Some people called it a t-shirt. Yeah. But it got called the hat and that's fine. There you go. On Jimmy Kimmel, at the end of his monologue, he did a joke about the hat and then he had his production team put a pretend tile hat on his own head. So one can see the hat on television on Jimmy Kimmel's head, if they like. Highbrow humor. Yes, very highbrow. So David was playing with this particular polychite that was to become the hat, and he realized that he was able to make larger and larger tilings with it, but they were not repeating. Ordinarily, you come across a periodic tiling fairly quickly. And with this case, he just kept coming up with larger and larger patterns.

He couldn't find a fundamental domain of repetition for a periodic tiling. And that's when he realized that he needed someone to help him. And that's when he brought in Craig Kaplan, who is a computer science mathematician, tiling guy, and a big supporter within the community that David was in. So David knew he'd have some software that was going to help. Wow. Now, this has been decades since the 60s. Do you remember hearing for the first time about the discovery? Were you surprised, suspicious, skeptical? No. So what happened, this is ridiculous. I was on social media and someone had posted a video that Craig made that shows the continuum of monotiles. So I saw this cool video. It was entitled

An A-Periodic Monotile. And I thought to myself, there's no A-Periodic Monotile. And then I clicked on it and immediately discovered that, yes, there is an A-Periodic Monotile. I immediately downloaded the paper. And I personally know a few of the authors. I knew it was going to be legitimate right away and I was very, very excited. As I understand, there were other shapes like the turtle and the specter, which are also built out of the kite construction. Do you think that there's an infinite number of these single tiles that can aperiodically cover the plane? It's really interesting. So in tiling

theory, one thing you can do is you can look at deformations of a tile. So if I take a square and I bulge out an edge and then make a corresponding tile with a bulged in edge that can fit with it, what properties of the tiling are preserved and what are lost and this sort of thing. So we look at deformations a lot. And so it's not surprising that you could make deformations that would still be aperiodic. What is surprising is that you can make deformations and still only have one tile shape and that that tile shape is also a periodic. So in my little description of bulging out an edge of the square, that immediately makes me need two tile types. I need a bulged out square and a bulged in square for it to fit next to.

So a lot of times when you do a deformation, you change the number of tiles in the tile set. So it's doubly remarkable that we can deform it and it stays a monotile. And is it a continuous family? Like you can change a certain parameter and move between these sort of the hat, the turtle and the specter, let's call them. Yes. And what is really interesting is that in the continuum from the hat to the turtle, there's one special parameter right in the center where you got not a monotile. So one of the things that people didn't like about the hat tile when it came out is that technically you need it and its reflection in order to form a tiling of the plane. So yes, it's a monotile, but you couldn't just make one glazed tile and tile the plane with it.

You'd have to manufacture a second one. Yes. And so is that really one tile or is that kind of two tiles? Feels like kind of two to me. Just like a left-handed glove and a right-handed glove. They don't feel like one glove. Yes, exactly. You need both of them. And so David didn't like that there was this criticism. He wanted to really get a real monotile. And if you take the tile that's right in the center, in the continuum that we now call the specter, David realized that if you only allow it to be in one reflection, as if it was a glazed tile, then it is an aperiodic monotile. And so now we have

what I think everyone pretty much agrees is a proper aperiodic monotile in what's called the specter. Now, these discoveries are incredible. And as you've already mentioned, as what happened with Penrose, they can foreshadow a breakthrough scientifically. You mentioned the one in chemistry and crystallography, which eventually earned a Nobel Prize. Your work relates these tilings to quasicrystals. Can you tell me what a quasicrystal actually is and how that's connected to this idea of tilings and groups, how that's connected to this idea of tilings and groups, symmetry groups in particular. Yeah. So crystals are solids that have a highly ordered atomic structure. So you think about a diamond. The carbon atoms are in these perfect tetrahedrons that just repeat over and over and over again. And when you run a diffraction experiment on a crystal

by passing some kind of waves, light waves, electron waves, X-ray diffraction is also done. Whatever wave you've passed through the crystal is going to bounce off the atoms and combine with constructive or destructive interference. And when you capture the results on a screen, what you're going to find are sharp, bright spots of concentrated constructive interference on a black background where there's a lot of negative interference or destructive interference. And crystals were defined as matter that diffracts with these Bragg peaks, with these sharp bright spots. And that's because they're so geometric and so ordered? Right. So the bright spots, the constructive interference happens when you have a distance

between two atoms that happens over and over and over again, so that the constructive interference is reinforced over and over again. A standard diffraction experiment on a solid could reveal something about the atomic structure of the solid. And so symmetries that you're going to get on your diffraction image have to be consistent with the symmetries of the crystals themselves. And so in the 80s, when Dan Schechtman was in the lab working on what we now know as the discovery of quasicrystals, he was constructing an aluminum manganese alloy that he cooled. And when he put it in the diffraction experiment, he got these wonderful Bragg peaks, but they appeared in this five-fold symmetry pattern that's not allowed for periodic objects at all. And so on the one hand, his diffraction image said crystal because it had

all these bright spots that come from this repetitive presence of the same structures over and over again in your atomic structure. But on the other hand, it said not crystal because not periodic because crystallographic restriction was broken. And the results were incredibly controversial at the time. And he ran into a lot of resistance from the scientific community that he had just made a mistake and that this wasn't actually a real phenomenon. A lot of people have that experience with surprising discoveries, and maybe it's not entirely wrong that reproducibility is an important element in science, but it can also be a little bit disheartening. But he perseveres.

He does persevere, and he keeps telling people, look, it's not a mistake. He does persevere and he keeps telling people, look, it's not a mistake. I heard through the grapevine, I don't know if it's true, that he had a colleague come over with an elementary chemistry textbook and be like, see, it can't be. You're wrong. some other people to sign on with him and get this paper published. It took two years for the paper to become published, which in physics terms is forever. And after he published this, people went back and dug out all of the diffraction patterns that they had produced over the years that they just poo-pooed, but were in fact actually were evidence of quasicrystals from prior. Wow, that is a lesson. Are these quasicrystals naturally occurring phenomena or are they totally human-induced? Well, there is a meteorite in Russia that has quasicrystals in it, and that's the one that I know of.

I think we've made the rest of them. Mostly, we don't know how nature produces these on its own, or even if it wants to very often. Now, much of your work in particular is focused on these hierarchical tilings. Tell me a little bit more about how those come about and what I'm supposed to kind of imagine when I'm thinking of these hierarchical tilings. So you put them together and you're forced to replace it with a three by three array of these other squares. And then I'm going to grow that three by three into a nine by nine and then a 27 by 27. And you can kind of grow these tilings in ways that are a lot of fun. And they come out highly structured, but not periodic. And that makes them really good models for quasicrystals, because you can ask, within these rules of this game, how disordered can I get?

How much order can I get? And what are the statistics of the kinds of things that we can come up with. The way you're describing this reminds me of fractal sets, where you have a scaling symmetry often on the boundary of something. So if you're in 2D, what would be a line as the boundary of a region becomes a fractal as the complexity increases? And we no longer think of it as having the length of a single line, but it's also not a plane. It's somewhere between the two dimensions. Is there a relationship that's well understood between these kinds of hierarchical scalings and things like fractal sets? Definitely. So fractals, when we think about self-similarity there, we think about it going

down on an infinitesimal scale. So no matter where you look in the Mandelbrot set, you're going to be able to find little small Mandelbrot sets there. So self-similar tilings kind of go the other way. So you zoom out and see the same general structures over and over again. Often, the tiles that you end up with in a self-similar tiling have fractal boundaries because that was the only way they could be self-similar is to have the self-similar boundaries. That's really fascinating. I did want to ask, what are the big remaining questions? There was this kind of challenge to find a monotile, the Einstein tile. Is there a big challenge like that that lots of people are scrambling after at the moment?

I mean, there's not a great, easily statable problem like the Einstein problem. There are lots of very active fields and some key problems people are interested in, a lot having to do with diffraction and spectral theory and trying to understand the degree of disorder we can create within this highly ordered setting, and then also creating tools to analyze these structures. and also creating tools to analyze these structures. There's people working on problems introducing randomness and seeing what's possible there. I suppose there's one more big piece of monotile-related work that I didn't mention yet, which is the work that Terry Tao and Rachel Greenfeld have done on the periodic tiling conjecture. So in 2022, they produced an example of an aperiodic monotile in something like 2 to the 100 to the 100 dimensions. So very much not an aperiodic monotile of the plane,

but an aperiodic monotile of an incredibly high dimensional space. But what's interesting about that discovery is that it pushes our limits of symbolic logic and our idea of what's computable again. So back to that question of decidability for Wang tiles, given a set of Wang tiles, can you write an algorithm that will decide whether it can produce a tiling or not? Tao has mentioned that he thinks there's a possibility that there's an undecidable tile in the sense that you might be able to construct a tile that no one could ever prove tiles or prove doesn't tile. Wow. So hearkening all the way back to Gödel's incompleteness theorems and undecidability. Yes. I don't think anybody's working on that one except Terry Tao. Just too hard. Speaking of intimidating figures to be working on. Yes. The Mozart of mathematics,

he's being called. Just a little plug for my friend and co-host Steve Strogatz. Our first episode this season was Steve's interview with Terence Tao. So tune into that one if you missed it. Now, we've primarily been talking about 2D tilings, tabletops, floors, walls, and you just mentioned a crazy high dimensional space. Are there similar concepts that are applying in a more modest step up? Let's just say to three. in a more modest step up, let's just say to three. Yes, there is an aperiodic monotile in three dimensions, but it's very unsatisfying. It's basically just periodic layers that then corkscrew in a non-periodic way. Would it have been considered periodic if it was aperiodic two-dimensional? I guess it's kind of trivial just to stack the sheets. So you really want, we are not done yet in two dimensions. It's sort of

hard enough. And so a lot of the diffraction work and the spectral analysis and this sort of thing is still happening in the two-dimensional spaces, but people are looking beyond for sure. So just to clarify, when I think about topology, I think about the connectedness of a space, the geometry of a space. What do I think about topology, I think about the connectedness of a space, the geometry of a space. What do you mean by topology in this context? What's the topology of a tiling? Right. What you do is you make a space whose elements are infinite tilings of the plane. And you have to put a metric on that space. Sorry, just a metric is like a measure of distances, and that's how you measure the space. zeros and ones. You could decree that two sequences were close if they agreed on a large block of digits. And so it's actually the topology of those spaces that we look at.

We put a dynamical system on there, and we bring the tools of dynamical systems theory in to study the spectral theory, and then the topology is good for studying rates of convergence and this sort of thing. In my world, we look at tilings as representations of the universe. And periodic tilings, you're imagining that the universe is actually finite. And each time you leave one of the tiles and go into the next, you've actually wrapped around the whole universe. It's kind of like a game of Pac-Man. and go into the next, you've actually wrapped around the whole universe. It's kind of like a game of Pac-Man. You go out the left-hand side and you reappear on the right-hand side.

So it's a tile. And you can imagine by reproducing those tiles in a line, you have the whole game of Pac-Man. Or you can just exit the left hand and come in the right hand. They're the same. I want to close by asking a question we like to ask our guests here, and that is, what about your research brings you joy? For me, it's very visual. I love having ideas that become images that I can see and gain intuition from. And I write a lot of code to verify the ideas that I have in my mind. And then I also

produce a fair bit of artwork coming out of my research. And that's been something I've been able to share with my students, the synthesis of math and art. And that's been something that many of them have found really inspiring. I love making the images. You know I want to ask about your home decor, like how your bathroom floor is tiled, your shower wall. In the Hudson Valley region, there are three tiled bathroom floors, one in my house, one of my mom's, and one of my sister's. And so I designed a tiling out of squares that is a self-similar tiling. Actually, I designed three different ones, one for each house.

And we had them laid and they came out really pretty. I'll send you a picture of my son's bathroom floor. Oh, please do. And actually other mathematicians that I know have also tiled their bathroom floors with entertaining tiles. There's a hierarchical tiling called the pinwheel tiling that's easy to make. If you have one by two rectangles, you can slice them up and put them back together in a different way that's aperiodic. And my friend Lorenzo Sadoon has that in his bathroom. Right. We're all crazy. All of us are a little weird. I love it. I will send you pictures. They are very cool. I really look forward to seeing them. We've been speaking with Natalie Preeby-Frank, a mathematician at Vassar College. Natalie, what a fantastic story and such a pleasure to speak to you. Thanks so much for joining us. Thank you. Thanks for listening. We have images of some of these aperiodic tessellations and of Natalie's mathematically inspired tile floors up on our website at www.quantummagazine.org on the

transcript page for this episode. Head there to take a look. If you're enjoying The Joy of Why and you're not already subscribed, hit the subscribe or follow button where you're listening. You can also leave a review for the show. It helps people find this podcast. The Joy of Why is a podcast from Quantum Magazine, an editorially independent publication supported by the Simons Foundation. Funding decisions by the Simons Foundation have no Thank you. from Matt Karlstrom, Samuel Velasco, Nona Griffin, Arlene Santana, and Madison Goldberg. Our theme music is from APM Music. Julian Lin came up with the podcast name. The episode art is by Peter Greenwood, and our logo is by Jackie King and Christina Armitage. Special thanks to the Columbia Journalism School and B Bert Odom-Reed at the

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