The Shape of Tiles: Regular and Irregular, Hard and Soft - Alain Goriely

Tiling involves filling a plane or space with repeated elements, known as tiles. This simple concept is deeply embedded in the natural world and human design, appearing in structures as varied as the hexagonal wax cells of a beehive and decorative wallpapers. While regular hard tiles—geometric shapes with straight edges that fit together without gaps or overlaps—are common in human-made designs, nature often favours soft or irregular patterns, shaped by physical forces. In this lecture, I will explore how both regular and irregular tiling patterns, hard and soft, emerge in nature and the underlying mathematical principles that govern their formation.

This lecture was recorded by Alain Goriely on the 28th of April 2026

Alain Goriely is a mathematician with broad interests in mathematical methods, mechanics, sciences, and engineering. He is well known for his contributions to dynamical systems, mathematical biology, as well as fundamental and applied mechanics. He is particularly well known for the development of a mathematical theory of biological growth, culminating with his seminal monograph The Mathematics on Mechanics of Biological Growth (2017).

He received his PhD from the University of Brussels in 1994 where he became a lecturer. In 1996, he joined the University of Arizona where he established a research group within the renowned Program of Applied Mathematics. In 2010, he joined the University of Oxford as the inaugural Statutory Professor of Mathematical Modelling and fellow of St. Catherine’s College. He is currently the Director of the Oxford Centre for Industrial and Applied Mathematics.

In addition, Alain has enjoyed scientific outreach based on problems connected to his research, including tendril perversion in plants, twining plants, umbilical cord knotting, whip cracking, the shape of seashells, brain modelling, and he is the author of a Very Short Introduction to Applied Mathematics (2017). His work has been recognized by a Sloan Fellowship, a Royal Society Wolfson Research Award, the Cozzarelli Prize from the National Academy of Sciences and the Engineering Medal from the Society of Engineering Sciences. He was elected as a Fellow of the Royal Society in 2022.

The transcript of the lecture is available from the Gresham College website: https://www.gresham.ac.uk/watch-now/shape-tiles

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Transcript

SPEAKER_00 0:04

Let me introduce uh Professor Allan Gorielli to speak to us on the shape of tiles.

unknown 0:09

Thank you.

SPEAKER_00 0:16

Thank you so much, Sarah, and thank you all for coming. And for people joining us online, thank you also for joining. So, as part of the series of the geometry of nature, today I will talk about wonderful topics, the mathematics of tilings. So, but before I start, I want to acknowledge my good friend Gabor Domokos from Budapest, who had been working on this type of problems for many years, and a few years ago invited me to join in one of the projects, the one I'm going to talk about today. And his talk is mostly going to be based on two papers with co-authored with Akos, who is a professor of mathematics at the university. And Christina was a brilliant PhD student who has since graduated. And the work that I have here is mostly going to be based on this article from 2024. And the last part on a most recent one that was accepted only two weeks ago in the proceedings of the Royal Society. And also I want to acknowledge that I'm very grateful to them because they also share some of the brilliant visuals that they've been developing and that I will be using for this talk. So I'm going to talk about tiling, both in the plane and in space. And as usual, it's much easier to start talking about things in the plane. So let's start with plain R2D tiling. Here are some examples I'm not going to go through. You see, there are mosaic, there are patterns that you find in nature, there are patterns that you find in technology. So what exactly is a tiling? Well, here is another example, a beautiful example that I love because it's from the Alhambra in Granada. It's from the 13th century. So a tiling is a covering of a plane or space by shapes, and we're going to call them tiles or cells for these stokes. And the condition is that these different cells have to fit together exactly with no gap and no overlap. And you see the tiles, the cells in this case, can be made of polygonal shapes with straight edges, or with curved ones. We're going to consider both, and that's going to be important. So let me give you a few more definitions. You can have what we call monohedral tiling made of a single shape that covers the whole space, things, for instance, square. But here is a better example. There is a delightful little book by Okusai, the same Ocusai from the Great Wave. And in his book, he studies patents and tries different types of patents that do exactly what I told you, which are tiling made of single or multiple shape. In this case, we call it monohedral because the shape is polygonal shape. It's made of straight edge of straight faces, by opposition to monohedric tiling, which are tiles that are made with curve shape. And here is an example, a beautiful example recently discovered in 2024, which is called the spectrum. It's a monotile, so it is single tile cover the whole plane without ever a periodic repetition of any pattern. It's an aperiodic tile. It's quite wonderful because people had been looking for that for the last at least 50 years. And I brought a few of them here. If you want to come after, I invite you to join and look at this in more details. This is tiling that's done with a single cell, but you can do with multiple cell. You can do a polyhedral tiling, polygonal in the plane, polyhedral in general. And here is Roger Penrose on the Penrose tiling. So back in 2020, when he received his Nobel Prize for Physics, we were in lockdown, and Darrell, who's sitting here, asked me, said, Well, the Nobel Committee wants some picture. Can you take some picture of Roger? And so I went around, took some picture, they put it on the poster and all that. And we, the two of us, were the only one in his acceptance talk that we had to film at the time. The Penrose style is one of the things he's well known for. The Penrose style here, the one in front of the Andrew Wilde's Institute, Andrew Wilde's building, the Mathematical Institute in Oxford, it's made of two rhombite shapes that you can put together. And again, this is an aperiodic tiling that looks like that when you see it from the top. Very, very interesting. You can have a polyhedric tiling, for instance, from Hesser, where the shape tiles, different shape tiles the entire plane, but this time with curves. Okay, so here is another example from nature. This is cracks forming in lava that has not yet completely solidified. And we're going to take the ideal version of that, and the ideal version is a tiling, and you can imagine it goes on forever. And in this tiling, you have different entities. You have two different mathematical objects of importance. You have the cell and you have the node. And what we want to do is assign numbers to that in order to quantify and compare different patterns that we observe everywhere in nature. Before we do that, there is an important distinction about the cell. A cell can be convex or non-convex, that's a technical term. It's convex if the line that I make by choosing any two points in the cell is also in the cell. So in this case, it's in the cell, and the one on the right is non-convex because there are points for which the line is not entirely within the cell. Okay, so it turns out that the mathematics of convex cell is much easier because it puts more constraint, and so that's a big difference when you try to uh to try to establish general results. Okay, so what is a node exactly? A node is a place where you have at least two edges meeting. And now I look at the cell and I look at its boundary, its perimeter, and I can count the number of nodes that are around the cell. So I can assign to the cell that number, in this case, there are five nodes around, but there is an important distinction here between the difference between a node and a corner. Because if you have a node like the number five here, where the edges meet at a T shape, well, there are only four corners within the cell. So if I if I'm in the cell and I look around, I would only see four corners in this case, right? The shape would have four sides. And so I make a distinction between the number of nodes and the number of corners. It turns out we're going to mostly look at corners and trying to soften them in due time. Okay, so now we've assigned these numbers, these two numbers, uh on a cell. What can we say about the nodes? Nodes are always like dual from the cell. Well, you can look at one node and look how many cells there are around. Just like we look at a cell and how many nodes there are around, we look at a node and look how many cells there are around there. So for instance, in this node, there are exactly five cells around it. So we call it the degree of a node or the degree of a corner. In this case, there are only four corners meeting there. So let's look back at our little example. Here is an example of a regular node. There are three edges meeting, and here is an example of an irregular one. You see that there are three cells meeting at that node, but only two corners, because one is flat. Okay, big difference. And for the cell, this cell has five sides, it's a pentagon, but there are six nodes around it. Okay, and I'm mostly be interested in the V star, not the V. You'll see why in a second. Okay, so now I can consider any type of tiling, and I want to assign a single one of these numbers to the whole tiling. So what do I do? I pick a point, I look in a circle or a ball of radius r, and I count this I average, I average these symbols for each one, the degree and the number of corners. Then I take the limit as r goes to infinity, and I obtain a single number for that. So that gives me two numbers, and there is an extra one, which is the number, the proportion of regular nodes. So the total number of regular nodes divided by the total number of nodes for a radius, and I increase this radius and take the average again. So in T's case, for instance, there are 11 regular nodes, all the red ones, and there are four irregular ones. So P is 11 divided by 15, and P is a number between 0 to 1. It's 1 when all the nodes are regular, there is no T, and it's 0 when every node has a T shape. So out of that I get three numbers, V star, N star, and P. Okay, let's look at a few examples so we get a little bit use of what that means. The simpler one, of course, the square lattice. If I look at a node, each cell has four sides, and four edges meet at each node. So I have four and four. All nodes are regular, right? There is no T junction. And you find plenty of examples, and I can never resist giving examples from Herbert Hook, who is the West Gresham professor of geometry, whose name is on the side here of the room. And here is an example from Micrographia from 1665. Very interesting book where it does the first microscopy book. It's also the first international scientific bestseller ever. And in there it looks at tissue. And for instance, the cork is a very clear square lattice that he dissects and draws within there. It's also the book where the word cell is introduced, the same word that we later use in biology for a cell. Another example is the hexagonal lattice. And here you see that I have the shape of six edge, hexagon, so V star equals six for each cell, but at each node there are only three edges meeting. And you see that there is a nice proportion between the number of edges and the number of sides that I uh and the number of edges meeting and the number of sides. Meaning if I have a site, uh a cell with more sides, I'll have less room to fit them together. So that I have an inverse proportion between the two. In this case, all nodes are regular, and of course we have plenty of examples like the honeycomb, for instance. Okay, these two examples. Now I want to organize my finding, and what we do as mathematicians, we put that in the plane. We call it the symbolic plane. It's also called, we call it also the Schlafley plane from Ludwig Schlafley, a 19th-century mathematician who works on polytops. His work was completely ignored until recently. And what I have is on the x-axis, I have n, right, the degree of the note, and I have the number of corners V on the y-axis. And so I told you I can take any tilings that I have, like Alhambra or the one I show you, compute this number and put them there. And the first result is exactly the one that I told you. If I have a regular tiling, there is an inverse proportion between the two quantities. It makes that they are on hyperbola. Here is the 4-4, the one that we understand, very easy. Here is the one we saw, the hexagonal lattice, and the dual of this one is the triangular lattice. It has six and three, right? And all the other ones that are there, all the regular tiling, are on this hyperbola. And here is a couple examples. You can create plenty, you can populate everything around. But you can also go around and look uh and look at different tiling that you observe in nature. And for instance, if you look at uh fracture that you can find on rock due to motion and all that, what you see is that because of this shearing motion that you have in fracture, you have lines that create naturally a tiling. And we call that primitive tiling, a tiling that's made of random lines in the plane, like in this case. And I want to be able to say something about this primitive tiling, where if I take random lines in the plane, the probability that the line, two lines would join is one, right, at this intersection, and it's very unlikely that the the lines would three lines would meet at one point. That's probability zero. So automatically I know that n equals four. There are four cells around that node, and I can use this theorem to conclude that on average the number of sides of uh primitive tiling is four. I might have five, three, four, six, etc. But on average, it's going to be four for a random tiling. So that's one example that I have right away, and I can place that in the symbolic plane right there. Okay, now we understand something about regular tiling, and of course you want to know about irregular tiling. What about this irregular tiling? We know this from school, of course. So, what about irregular tiling? Well, there are also plenty of examples. Here is an example with made of two squares, the yellow and the red, and you see that there are three edges meeting at each node, all the nodes are the same, but there are only two corners, right? Because of this T section. There is the corner from the red one below and the yellow one below, but not at the top one. So n star equals two. And all nodes are irregular. In every single node, there is a T section, a T junction. So P equals zero in this case, and that's how we put bricks, and that's our cell, for instance, tissue in the in plant tissue also organized. There's plenty of good reason to do that because that creates a stronger structure. So where do they sit, the completely irregular tiling? Well, more or less they they follow more or less the same kind of law, meaning that there is an inverse relationship between the degree and the number of edges, and they sit all right there. Okay? So now we have started having a good understanding. What about everything between irregular and regular tiling, where all convex polygonal tiling sits in the symbolic plane in that little zone here? Right? And now we have a complete classification, and we can go around, and that was Gabor and his group did for many years. They went around, took picture and computed things and computed his average and look at the different different examples, and they all fit somewhere there. So they've really, everything that you find, polygonal sit somewhere in this plane. And this is also the start of our conversation of what I want to tell you about today. The important message is okay, we understand that, but is there something else? Well, we know of other types of tiling, hyperbolic tiling. We know that there are non-convex polygonal tiling here, but what we were interested in is are the tiles with fewer corners, softer corners. Well, we know that polygonal tiling, the minimum number is three. You can't have a tile, a polygonal tile with two sides, right? So if you want something there, you'll have to curve things. Is there something there and how do we characterize them? And this is in a way the new thing that we've been working on, is the notion of soft tiling. It's not so new. Again, if you go back to Okusai, 1824, you look at some of his design. And what you see here, if you look at this shape here, each shape looks like a little banana like that, and this has only two corners. They're all different size, but they all fit, and you have a finite number of them, and you can tile the whole plane like that. So this is what we call a soft polyhedric tiling. Next question is can you tile the plane with a single cell, a single soft cell with two corners? Oh well, let's go back to hook micrographia. This is a drawing of tissue of seaweed. And if you forget about the little bobs, what you see is a cell here that repeats itself. And if you apply the same law that I give you, if you look at one node, there is at one corner, there is only one corner at each node, and there are two corners together. So I have V star equals two and N star equals one. If I make them a little bit straighter and put them together, they start looking like this shape here. Okay? Another way to think about that is I can start with a cell that I know that tiles the plane, and I can soften it by moving the top edges up and the bottom one up also at the same time so that they still fit together. Right? So I go from four corners and four edge to two corners and two edge. And I then I can put them all together like that and fit the plane. Okay, here is another example. You can go to the wine store, you can buy a nice bottle, and they'll give you the wrapping. You throw away the bottle and keep the wrapping. And what you'll see is that T's are also naturally soft cells. Each of T's tiles can tile the plane just by T lenticular shape. And if you look at one node, each cell has two corners, and each node has two corners meeting. So in this case, I have V star equals two and star equals two. Can you do better? Well, that's the result of the next theorem that tells you that the minimum number, the minimum number is actually two. You cannot have fewer than two corners for shapes that that will uh tile tile the entire plane for single shape. Okay, so this is this is uh a bond that we have. But now that you think about softening a given lattice, you can start with any lattice that you have and a straight one and say, Oh, am I going to soften this? Well, it's it's not soft because at the edge you have corners. So the way to do it is to move a little bit the corner so that the tangent at a node get becomes straight. So what you take, take a node, and what we're going to do is bend it and then do the same at every node, and then look at the other type of nodes and bend it again, and now you've removed all of them like that. And then you have a nice shape. So there is a there is a direct way to go from a given lattice that you want and soften it to one with fewer corners. And it's it's not that complicated here, but we're going to apply the same process when we get to 3D. And now you can take the three types of regular lattice that we know in the plane, the triangular, the square, and the hexagonal one, and you can soften them into this different type of shape, and we have a complete classification of this shape. So we call a soft tiling something made of soft cell. It's a soft cell if it has the minimum number of corners. And in the plane, that's two. A soft tiling is made of soft cells. And again, you look around, it's quite amazing because mathematicians for thousands of years have been thinking of fitting the plane with a straight line and all that. But you go look around and you find plenty of shape in nature, like in rivers and all that. And when you look at shape, they're all like this soft. Here below is uh the shape of a fungus as it grows. You have uh epithelial cells, the way they order themselves. Uh here's a piece of wheat. Uh blood cells naturally take this shape as they go faster and faster in the capillaries, but also in uh architecture. For instance, the work of Zaha Hadith is clearly not inspired by this idea, but connect directly in making this more organic shape with fewer corners and fewer straight edges that you find. You'll find them, of course, in the work of Escher, because he's been looking at all possible ways of tiling the plane, and he this is one of the possible ways to do so as he explored there. Completely intuitively, of course. Okay, so now we have a decent understanding of what's going on in the plane, about this of shape. I want to go and start talking about uh 3D space. And of course, things are right away much, much more complicated. So, for one thing, about uh 3D space, in the plane, every regular polygon exists. You can have a two-sided, three-sided, four side, no, not two sides, three sides, four sides, five sides, and sides. But we know since Plato that there are only uh five regular uh polyhedra, and T's are the one, and Plato had his idea that they can they make the constituent of all the different possibilities air, fire, water, uh, the universe, uh, earth, and so on. And the cube, for instance, is the constituent of all earth. In a way, you can think of that it Was thinking of atoms, he was thinking of things filling things and so on. So I think the basic idea of the tiling, that something is only made of one shape, was already there. But then came Aristotle and says, wait, wait a second. There is something that's not right here because these three shapes, they're regular and all that, but they don't fill space. You can put them together. There are only these two shapes that can fill space, which is the cube and the regular tetrahedron. And okay, the cube is quite clear. If I take a cube here, this is a Rubik's cube, another Hungarian invention. So if I take a Rubik cube, you say I have four cubes here, and if I put four more here, I would fill it, and at one node, I have the degree of one node is eight. I have eight cubes surrounding every node. So you can ask the same question about regular tetrahedra. How many tetrahedra do I need to put together in space to fill it completely? And for once I'm going to ask the audience for their opinion. A little vote here. Somebody wants to venture a number. How many tetrahedra do you put so that you fit you fit them completely? Like we know hexagon, you need three. Here for cube you need eight cubes. Anyone wants to try? I will I will uh repeat that for the audience over there. Well, I have sixteen here, let's do an auction. Price is right, come on. 12, I have 16, 18. 36. They're all good numbers. But the re the reality is that if you look at how many you need, it's 22.79. The regular tetrahedron does not fill space. And that was only understood later. Of course, there are plenty of other tetrahedra that fill space, like I have these magnetic toys here. It's all made of uh tetrahedra, as you can say. They're not regular, but you can directly demonstrate that they fill space. I put them in a cube and then I can have as many cubes as I want. There is a big space of tetrahedra filling space, but not the regular one. However, let's go back to our friend the octahedron. Here is the octahedron. Aristotle knew it didn't fill space, but what you can do is the following: you can truncate it, you can truncate the edge here, and to obtain the truncated octahedron, with an example of Archimedian's polyhedron. And in that case, you can put them together. And that's a very important one. It was studied by Fedorov, the mathematician, and Kelvin, the physicist, the mathematician, uh, in at the end of the 19th century, and we'll go back a few times to that. There are other shapes, the so-called parallelohedra tilings. These are the five possible shapes that can fill space by pure translation. You cannot rotate them, you just spike them like cubes if you want. Okay. These are the only the five possible shapes that you can do. Of course, you can change the aspect ratio, some of them, but the uh in terms of geometry, these are the only possible shape, and you see in the middle of front the truncated octahedron, and the cube is part of the parallel pipette family, of course. Okay, so plenty of work, thousands of work, of years of work on different uh polyhedra and tilings and all that. So, what's the angle here? Well, the angle is the same as before. Is there a version of this polyhedra that's softer than the regular, the one with straight faces and edge? So remember we started in 2D, we had four corners and four edges, and then we reduced that to two corners and two edges. So uh question for you is I take the cube, okay, let's count it as eight corners, twelve edge, six faces. Is there a way to move the edge, change the shape of the faces so that I can soften it, reduce, reduce the number of corners of that shape to make it a little bit more organic? And how far can I lower it, or far can I go? What is the soft cell in 3D? How many corners? Okay, so back to the option. I take any numbers. Okay, very good. It's already good progress. We went from eight to four. Can somebody should be four? You don't have to raise your hand. It's not it's not so how many how many corners would that give me at the end of the day? Oh, maybe not, that's interesting. Or can we do that? That's maybe a little pushing it. Okay. And congratulations, the price is right. The minimum number of corners is zero. There's so much space in space that you can deform things to remove all corners. And I give you a theorem, and you say, okay, I want a proof. This is not a one-like proof, this is a one-image proof, because this is the proof. This shape here that I have here, right? A 3D printed. And if you go along, you see that it has only one edge, it has no corners and only two faces. Okay, that's the first part of the proof. There is a shape like that that exists. The second part is that it has to tile the entire space. Okay, here the second part of the proof is that I can take two of these shapes, they have the same complementary, the same shape actually, then that is symmetric. I can put them together, and when I put them together, I make a long prism. I can put as many as I want, that has a square section, and then I can pile the prism in the other direction as I want. And that's the end of the proof. Right? It's quite remarkable. But of course, this is just the start of the idea. You know, this object exists and the minimum is zero, a soft cell is zero. How do you do that in general? Okay, so again, the idea is edge-bending. You look, for instance, going back to the cube uh lattice. If you take a cube lattice and identify one node, so you pick one node, what you must do is make this uh make this tangent disappear so that they go smoothly together, so that they align with each other. Okay, so if I just focus on that and follow the colors, what I have now, I'm going to start moving these edges and moving the one below as I must do. I do that on one node, so I have to do it everywhere else because I have a lattice. And then I extract that one cell, and now I have a soft cell here. And since I've I can wiggle things around without changing the number of nodes, I can transform into T shape, and then when I have T shape, I can color it and pile it on top of each other. And so this edge bending process can be done systematically to any kind of given lattice. You can soften any kind of lattice. So it's quite interesting because to see the progress of things are going in mathematics or science in general. We two years ago we had these ideas that uh you could do that for any lattice with a few very minor technical conditions, and so we put that as a conjecture. We had an exact test of how to do it, uh uh, that a certain condition had to be satisfied. Uh, and then we published that. And this morning I receive an email by two mathematicians and says, Oh, we really enjoy your paper, and we've been working on it. Uh, and we have this new paper that I'm sending to you, where we prove that your conjecture is actually right, and generalize it, of course, because they're pure mathematicians and they like to generalize things. But they actually prove it completely that you can start with any given known lattice and systematically apply this process of bending edge to soften and remove, remove the uh the corners. And here is an example of favorite example, the truncated octahedron. If I look at the wireframe, I can put it together and tile it, and tie the whole space like that. Now I can apply, I can take uh uh I can take an octahedron, truncate it, apply the edge-bending algorithm, and get T shape. And T shape is is super nice. It has it has no corners, it has very nice, very nice uh uh faces that we chose to be as minimal surfaces. So if you look at the edges here, the surface that we decided to put on top of it is the one that minimized that surface, uh the surface area for that given boundary. And so the wireframe looks like that, and you can put it together like that. So you can now start with any shape and apply systematically this step to create a softer version of that. You can go look around in architecture, for instance, and you see some of these IDs also expressed in different different uh shapes and different instances. But a much more interesting uh project application is something that we started talking with uh people, designers at California College of the Arts, and that's the group of Margaret Iqueda, Nega Kalanta, and Van Jones. And I've been we've been working with them for the last two years with different ideas, and that was one of the first ideas to use this shape as building block. Not only as building block, but this shape actually made of eggshells. Uh so they went around San Francisco and asked all the diner during the morning breakfast, can you give us the eggshell? They cleaned it, they ground the uh the shell, and then they added gelatin and some uh uh bacteria to make it all together. So these are completely uh uh bio in any way sustainable, uh, and they have extremely nice property as as natural building block. And as a result, they have a lot of different applications and they want a biodesign uh challenge based on that. A wonderful work that they've done, really beautiful. So, t-shape by itself, the fact that it's a different shape, gives not only ideas but new possibilities. And in particular, they tested the strength of it, and that shows that it has better mechanical properties than a regular brick. But the application that I particularly like, and that's how we started talking with uh with Gabor, you can find it deep in the ocean. So, people who came for my uh third lecture in Gresham about seashell, you've you've learned all about Nautilus, the iconic shape that has a logarithmic spiral, and we've talked about different properties of that. But today I want to talk about another property. It has a logarithmic spiral, but it has also all this chamber that you see. That's a characteristic of cephalopods. And these chambers are used by the animal to regulate buoyancy and go up and down so that they can come up from deep in the sea during the night in order to feed. There's a little opening called the cefuncal, there where it can regulate different types of gas and with that change the orientation and go up and down. Beautiful, beautiful physics. But the interesting thing is what is the shape of these? And you see that they have that the lens shape, the same one that you find in Ocucy, but the interesting part is the shape in three dimensions. And here is a CT scan of that shape. So now we see the shape in three dimensions, what it looks like, and you can extract it. It's a beautiful change. So it is one chamber within the nautilus. And you can extract it, extract that one chamber, that's what it looks like. And again, you can change a little bit the shape without changing the corners or anything like that, and we change it up to a shape that looks like this one. And this one is a soft-cell. So the nautilus chambers, that's it, create its chambers for reasons that we partially understand, create naturally this soft cell. It doesn't have any of these uh corners that are difficult to maintain, but can also be uh mechanically compromising for the overall uh strength of the material. So it's a very natural shape, and then you can either use do sprism or increase the size with that same logarithmic rule and feed the whole space like that, and you can see that it's an example of a soft cell. There are other examples, but in order to do that, I have to tell you about bubbles, uh, soap bubbles. So you can if you blow a soap bubble, naturally, without disturbance, if you it's going to take the shape of a sphere, and that's a balance between the inner pressure and the surface tension that try to minimize the surface area. And we told already about this minimization of surface area here. So a shape by itself, uh a soap bubble will be naturally uh spherical, but when you put them together, you have the role of the pressure, but you also have the role of the contact line balancing mechanical force. And in the 19th century, Kelvin, Lord Kelvin, tried to understand form, soap, soap film, and form in general, and tried to see if there was a simple model for it. And his first idea was to use the truncated octahedron, but he realized very quickly that it doesn't fit the basic rule of mechanical equilibrium at one note. Because the at one note, when you have four different edges meeting, you must have the same angle in all the faces. And the truncated octahedron as this 120, 120, and 90 degrees below for the square. And he says, okay, I can change a little bit, just like we do, I can soften a little bit. These are not soft cells, but they're softer than one because they're not straight edge. I can soften the edge and I can soften the faces to make what we call now a Kelvin cells, which has the same angle, about 109 in each, all the way around, the three angles, and that that satisfies the physical law and also can fit together. And up to this day, it is one shape, the Kelvin cells, is the one that is conjectured to have the minimal surface area for given volume as a single shape that tile space. It's still an open conjecture, actually. That's the first interesting uh thing about a soap film. It's almost a soft cell, which it goes into softening, but it's not quite there. The other thing is that it's also directly related to minimal surfaces. If you fix the boundary, like two circles, what you obtain in a soap film is a catenoid. It's the minimal surface, it's the minimal area for the given shape, for a given surface that has the two circles as boundaries. And you can, as you see in this picture, you can make it much more complicated. And why I'm telling you that, because Gabor went further, he says, okay, I can take the wireframe of the soft truncated octahedron, dip it in the soap, and the shape of the film on the faces will naturally take the shape of the soft cell. And so all you need to do is dip it in the in the soap. Well, it's very hard to do with everything, but it takes a long time. But in principle, it naturally adopts its shape just by giving the edges around. Okay, we'll we'll get back to soap uh to soap film in a second. So soon after we published the different result that I just talked about, essentially, we realized that there was an interesting connection with a completely different mathematical object uh that's very well known in geometry, which I call triple periodic minimal surfaces, and here is an example of them. I'm gonna tell you a little bit more about that. But before I do, let's go back to our friend, the truncated and the soft truncated octahedron. And let's look at a bit a little bit differently. So we started with the truncated octahedron and we soften it. And so how do we do that? Well, we actually don't change the position of the node. What we do is change the tangent at every node. So the tangent here, or the half-tangent, is horizontal. Every node is the same, so I can only look at one, it's a very highly symmetric object. So I have the cage there dotted, and to go from the truncated octahedron to its soft version, all I really did is change a tangent. That was the edge bending, so that now the tangent gets smooth from one to the next one, right? And so that's what I can do. You can see here I have two more spaces here. So the question is what else is in that picture? Well, you can bend the edge a different way, instead of going up, you can go down and create another soft cell, surprisingly. And this soft cell looks like this. It looks like it has a little piece of that. And what's remarkable of that is that this soft cell is the unit cell of the Schwarz primitive surface. So now I really have to tell you about this triply periodic minimal surface. So you can divide three-dimensional space with a plane so that you can have yellow on one side and blue on the other side. Or you can have a surface like this one that splits the same space in outside and inside, or blue or yellow, with this intricate uh minimal surface. And it's a minimal surface because it has the same characteristic as the one before. It's triply periodic because I can take one unit cell and translate it periodically in the three directions to create the whole thing. And that unit cell is a soft cell on its own. The Schwarzby surface, said the end of the 19th century, uh, was the simplest one, and really Ts get very hard to understand. So the complement of that on the yellow side is also a Schwarzby surface. So it's called uh double labyrinth, and the name really evoked, because they're quite hard to picture. This is the only one you can really easily picture. So you have a double labyrinth because you have two of T's structures sitting on top of each other. And if you make semin-blocked, it would look like that. So the you see here the first floor is only connected to the third floor, and there is no connection in the second floor. That's why you have two labyrinth. Okay? And these shapes, remarkably, they're not just a mathematical oddity. It turns out they're fundamental in many different aspects of science. In biology, they appear because fluid membrane tends to have these uh minimizer uh properties. And so cellular shapes often take the shape of minimal surfaces. And it turns out, for instance, in the description of the coronavirus, this type of model are used to describe the different shape of with negative curvature and so on. In uh in uh engineering, people use the T as microstructure because they have a lot of interesting properties in terms of thermal property, mechanical property. They have photonic band gaps, so they have different optical properties and so on, so they're extremely used in that regard also. And now they are a typical type of design for microstructure in a lot of what we call smart material. This is just a primitive one, and I've left a space in between, so I have the hard trunk truncated octahedron, I have the soft version, I have another soft cell on the other side, and all I did was I have a family that I obtained just by changing the tangent. And the question is what's in between? And lo and behold, what's in between? It's the Kelvin cell. There is a particular angle that makes the Kelvin cell. And now you see all this different structure: the hard one, the soft one, another soft one related to P-surface, and the Kelvin cell related to soap bubble, all coming to as part of one big family that is connected by this transformation of edge bending. Really remarkable. And that's just for the Schwarz-P-surface. Of course, now you can look at other, there are entire family of triple periodic minimal surfaces. And what the Schwartz was in 1865 or something, and described the Schwarz D surface. In 1970, Alan Schoen discovered other ones, the gyroid, which is even more complicated. And what you see is the unit cell themselves are uh soft cells also. And you see the gyre of the Schwarzzy surface, just trying to understand the labyrinth that goes in and out is quite difficult. And that's why also, because that's a labyrinth that connects two at the interface, you can have a lot of nice property if you have two liquid and something like that. So if you had oil in the blue side and vinegar on the uh yellow side, you would never be able to make vinaigrette, right? They live completely separately. But if the interface is made with the right property, you can have a lot of exchange there. And that's what engineers use for that. An example of that is the idea, a so-called uh dye block copolymer. So these are macromolecules that compose of two distinct polymer change. So when you mix them, naturally they would separate, but they are also covalently bonded, which is quite energetically difficult to pull them apart. So they want to be apart, but they want to remain connected. And the balance of the two is to make this again very complicated labyrinth that's not a simple uh triple periodic minimal surface, but uh have components of different ones. And in this case, you can think of these unit cells, these soft cells, uh mesoatoms for this uh polymer. And you can see that you can really do like uh Lego and bricks put them brick together to create this more complicated shape. So they're quite remarkable on a lot of respects. And we found now entire families of them related, which are different than the typical families that have been explored in mathematics so far. So the last thing that I want to talk about, we we went deep in the sea, and what better place to do space filling than to do space filling in space, of course. So uh when we published the first paper, it had a little bit of press and all that, and uh my friend Gabor managed to convince the Hungarian Space Agency to say, why don't you take one of these nice shapes to space? I say, yeah, but what are we going to do in space? But ah, here is an interesting experiment. Why don't you just take the wireframe of the soft truncated octahedron, the one in yellow, and fill it with water? So in space, no gravitation, no gravity. So if I have a uh if I have water, it would naturally take a spherical shape. Except that if I have constrained at the boundary, if I put like a circle, it will try to attach the circle by to uh to reduce its surface tension at the interface. Right? So if I have a given shape and fill it with water, what I would have is that if I have just the right amount of water, the shape itself, the entire volume, would take the shape of a soft cell. And that's what it did. So in July 2025, last summer, they went to the space station and started to fill with water that and you see it's quite beautiful with all the light and things like that. And as I say, it's the first instead of a 3D shape in negative surface curvature in space, okay, realized from liquid water. I'm not exactly sure it's you know it's not going to change the world, but it made this astronaut really happy. You know, and that by itself is something. No, you can say and it's so cool, right? Shape in space. Okay, so uh let me conclude with a few points here. So what we've seen is that there is a systematic way to soften this shape, and you can say what the point. It turns out that what we systematically found since then, or and that what we had before, just like with the Nautilus and other shape, is that there are many shapes in nature that naturally adopt these soft shapes, and that they are better descriptor of what we saw, what that what we see rather than the traditional uh uh uh flat ones. Even if you look at the honeycomb, you realize that if you zoom enough, well, the honeycomb is not perfect. It is shape actually around at the edge, and if you think of the space in between, it is actually soft cells also. It turns out that it takes an extraordinary amount of energy and organization, and typically extra protein, to create polygonal tiling within biology. And for 3D cell, we have a partial classification of some of the shape, and now I haven't read the detail of the paper, it's still not peer review, but according to uh the mathematician, you can do it completely systematically. And there are plenty of other types of uh soft cell, like the tri associated with the triple periodic minimal surface. You can have uh carbon allotropes that form this Schwarzsi that are also part of these uh uh properties that have the same geometry. You can have various types of microstructure, but one of the nicest ones is the one that you find on uh butterfly wing, and here is an example of a weevil. You wonder why they're so so glittering. It turns out that the shape, the the type of structure that you have right on the wing or the carapace of T are made of different layers that organize themselves very closely to triple periodic minimal surface that create optically photonic gap that creates all this glittering aspects. And so there is a natural way that nature used that for uh for optical purpose, in particular the gyroids. And probably uh most importantly, uh, when we were done with that and we received a little bit of press, uh Scientific American called us and told us that it was the coolest mathematical discovery of 2024. We were top one on the list. And so that's one for the the bucket list, I guess. So again, thank you very much, and I want again to uh thank my collaborator Gabor, Akosh, and Christina for the