 Hello everybody and greetings from Northampton, Massachusetts. I wish I could be with you today. I'm so glad that IHES is celebrating Louis' 100th birthday. It's almost as if he were there to enjoy this. I'm going to share my screen and tell you the story of how I met Louis and what I would give him today if I could. And I hope you can see this. So the title of my talk is A Gift for Louis, the new A Periodic Monitile. But I want to begin with telling you how I met Louis and that was not at the Ottawa Congress, but as a result of it. So the International Congress of Crystallographers in Ottawa in 1981 was for me a turning point because I met a young medaller just named Denny Gracius. And I presented a poster on color symmetry and his poster was next to mine. And when we had off times, we read each other's. And so Denny read mine and he said, ah, you have to meet Louis Michel. And I was nervous about that. I told him I had tried to read Louis' papers on symmetry and physics. But he wrote in a formal abstract, take no prisoners mathematical language. And I am a mathematician, knowledgeable about symmetry, could not get through the first paragraphs. So I was afraid of Louis and didn't want to meet him. But Denny laughed. And he said, no, no, he's nothing like that in person. So when I got home, I remember that I belong to Smith College, where I taught mathematics, belongs to a five college consortium. And the five colleges committee on applied mathematics had funds for visiting lecturers. And I was a member of that committee. So I proposed that we invite Louis Michel to be a visiting lecturer there. And my suggestion was accepted immediately. It was a committee of people from five different colleges, two maybe or so from each. And everybody was very enthusiastic, because they all knew of Louis. They knew of his broad interest, his expertise, and reputation. And everyone had a stake in the visit. So we agreed to invite him and we did. And Louis was appointed distinguished visiting professor of mathematical sciences at the five colleges. And this was a wonderful thing for all of us. And we're grateful to this day that he was here. And that is how our friendship began. So Denny Grassage was right. Louis the person was not the distant austere formal author of his papers. He, Louis the person was funny. He was lively. He was informal. And he was interested in everything. And so was Therese, his wife. And it was wonderful to get to know them, to have them there. They visited us many, many times. Louis also knew what he didn't know. And he was eager to learn from anyone who could teach him. And I think that's why he took an interest in us. So he, in addition to visiting us, he invited each of us to IHES. And that was wonderful experiences for all of us. He had worked with Niels Bohr and Copenhagen. And he and Therese were deeply impressed by the Bohr's hospitality. And the close relations among colleagues that they fostered and vowed to make their home and bureau a welcoming place. And they certainly succeeded. And the Michelle's dinner parties, which seemed to be every night, it was not unusual to speak one language with the person on your left, another with the person on your right, and the third with the person across the table. Therese was multilingual, ever alert and rescued her befuddled guests. She spoke French, of course, English, of course, and Chinese and Hebrew, and I think a couple of other languages. So we were always saved by Therese. This is a picture in their kitchen, where the meals were usually served. The usual gift to bring Louis, the gift that he most enjoyed, was an example of some curious phenomenon in mathematics and physics, or a new result or a novel proof. So though it isn't really mine to give, my gift for Louis on his 100th birthday would be the curious, novel, and very new, 13-sided polygonal, a periodic monotile, also called a hat, and an Einstein. Einstein meaning single stone, if they should be two words there in German. This is what it looks like. That's a very strange looking shape, but it does have 13 sides if you count them. And it's a polygon and believe it or not, it's a tile. What would Louis do with the a periodic hat? Would he wear it? People did apparently wear hats like that back in the day. Would he eat it? Two days after this hat, tiling was announced, cookie, the code for 3D printing was made available to everybody worldwide, and people made cookie cutters and began to cut the cookies out of the tiles and eat them. So you can play with these and then eat them. Let's go back. And then the other possibility, which I'm sure Louis would do, is to set the tile in context. What is this, what is the special case of how can we understand it better? And here, that's what we did for several years. Peter Engel and I and Louis worked together on a project we call Lattice Geometry and we were going to publish his book. We never were able to do that by the time Louis died and fortunately for all of us, Boris Zdolinski has done much of that and he's already spoken to you about this. But there are things in the preprint that are not in his book and some of those I think would be relevant to this. So Louis always asks, what is this a special case of? When he learned that, for example, these are some examples that are in the Lattice Geometry book, when he learned that Conway and Sloan found an 11 dimensional lattice generated by a set S of shortest vectors, but with no lattice basis in S, he proved that this occurs in every dimension greater than 10. So this was just one of the first in a large infinite family, which Conway and Sloan themselves had not realized. When Louis learned that he had conjectured on Lattice Geometry that we'd been trying to prove for several years, I could say, failed in dimension eight, he generalized it and defined new lattice invariants. So this was his way with a curious novel phenomenon was to ask, what's a special case of and find that special case? So let's take a closer look at the hat. And we see that it is divided, it can be divided into eight pieces, congruent to each other, and these are called kites by some people. So this is an octokite because they're eight of them. And where do they come from? They come from the hexagonal tiling of the plane except that in this case, we partition hexagon, not by drawing radii from the center to the vertices, but to the midpoints of the edges. And then we partition each hexagon into six kites. And these particular eight are the configuration, joining together the configuration called the hat. And here is the way that we appeared in the New York Times, which gave it a full page. There's been so much excitement about this creature. And here you see the piece of the larger piece of the tiling. All of these shapes are hats. Some of the dark blue are mirror images of the other, but that is debatable whether some people don't like that, other people don't care. Anyway, they're all hats if you accept mirror images. And this is all situated in the hexagonal tiling so you can see how it fits. And a reporter asked me about that. And I told her I was just amazed that it was sitting right in the hexagons. He would think that an aperiodic tile would be a very strange, weird thing that never had anything to do with classical crystallography or classical tiling theory. But here it is. It's just sitting right there. This is very, very new, as I said. The primary source is so far, March 20th this year. David Smith, Joseph Myers, Craig Kaplan and Chaim Goodman Strauss posted a paper they called an aperiodic monotile on archive. And it went viral. And then a few weeks later, May 2nd, Joshua Sokolar who has been involved in tiling theory for a very long time, physicist at Duke University have posted, not published, quasi-crystalline structure of the Smith this being the same Smith monotile tiling also on archive. And then there is also a lot of material found now on various websites. But nothing has been published in the peer review journal yet. However, it's being reviewed by peers all over the world. And no one so far has found anything to object. So this is Smith, Myers, Kaplan and Goodman Strauss. What do they do? They first prove that the hat tiles the plane. That you have to do that. That's not obvious. And if you cut these out, copies of the tiles and play with them as I did, you find that you can begin to fill the plane. But you can't, there's no guarantee, no obvious guarantee that you can continue that and fill the entire plane. And there are, of course, we know many tiles that you can tile finite regions, but yet still can't continue that. So there's no reason to believe that, but nevertheless, they were able to prove that. And then they give two different proofs that the tilings are aperiodic. One proof is modeled on the proof of aperiodicity for Penrose tilings, though it differs in some respects. And the second proof seems unlike any used before. So the Penrose tiles, I think you all have seen these and you know them very well. These, they're the two, I'm taking the, they come in several different forms, but the form I'm using here are two rhombuses. This was 72 degree angle here, pi over, two pi over five, here's two pi over 10. And the notches force you to, you have to, they're called matching rules. They force you to assemble these in only a certain number of ways. And in particular, you cannot assemble them into a periodic pattern. So what you can do around single vertex do five of the so-called thick realms in two different ways and so forth and so on. And if you examine a patch of the Penrose tiling, the matching rules aren't shown here, but you can examine all the vertices and see that they are in fact, always one of these various, these eight. So that's what Penrose tiling is. And they've had tremendous applications and inspirations in many fields, including in crystallography. The thing about Penrose tiles how they were able to prove that they are in fact non periodic is that in every Penrose tiling, we can group the realms into larger realms, as shown here, the red ones. And this is, and there's only one way to do this. In other words, they have a hierarchical structure that Penrose tiling then contains within itself another Penrose tiling on a larger scale. And those larger, that larger scale tiling does the same thing. It contains another Penrose tiling on a scale larger scale. And I didn't continue to draw this in, but you could imagine how that goes, just like the red one. And so you have a blue, a larger scale. And then you can take that, of course, and do that again. And this goes on ad infinitum. And then from there, that's a not too difficult argument that the Penrose tiling is not periodic. And Smith and his colleagues showed that this is the same thing for the hats. And it's more sophisticated, a little differently, handle a little differently. But they showed that although hats cannot be grouped into bigger hats, they can be grouped uniquely into four different metatiles. And the metatiles here are shown here. They're four different polygonal shapes. And the hats fit into those more or less. So all of this is very much like the Penrose prove not quite, but enough that it's convincing. The metatiles can be grouped uniquely into metametatiles, not exactly the same shape, but of the same basic shape. And this converges just as in the other case. And so therefore, the hat is a periodic too. So the paper, the original paper is 89 pages. This takes up half of it, but I won't go into details here. Joshua Socolar, more recently and by a few weeks after their recast, the hat tiling's in the high, high dimensional cut and project framework that had been developed for Penrose and other a periodic tiles. And he's calculated their diffraction patterns. And so here, his version of the metatiles is shown here. And this is a picture of metatiling. And here is a diffraction pattern that Socolar has calculated. And you see it's hexagonal. It's not, it's a classical crystallographic. But it is also, he's able to show that this is, in fact, not periodic. The second proof, uses a process similar to zone contraction of zoned top families. And this is a classification of lattices that we studied in lattice geometry and Louis was very interested in it. But this doesn't fit that exactly because the hat is not a zoned agon. Nevertheless, some of the ideas are similar. The hat has eight edges of eight length one. Those are the brown edges shown here. This one, the longest one here, we think of in this case is two short edges, the equal length of these. These are all length one, one, one, one, one, one, one, one, one, one, one. And it has six of length square root three. And so what they do in the second proof is to study the tiling as these different edges shrink to zero and then expand back again. If you shrink the brown edges to zero, the tile contracts to this, which is a hexagon. If you shrink the green edges to zero, the tile contracts to this, which is an octagon. And every tiling by the hat belongs to, this is what they show, a continuum of tiling that includes one by tetriamons. These are tetriamons, meaning four equilateral triangles and one by octiomons, which are eight equilateral triangles. But these are smaller than those. The area of this one, which is four, the four is actually three halves the area of this one. And this is a crucial point for their proof. So let's take a look at this process of the, this is Craig Kaplan's video or animation. So you see the edges are shrinking. And then when it completes, it's shrinking. The green ones are gone now. We're in the shape. This is the shape of the octiomon. And now it's, we're letting those increase again. The edges come back. Somewhere along in the middle here, you get the hat. As we continue this edge shrinking, we move over to the browns being zero. And now we have just the green edges, which are, we get this tiling. So let's see if we can, aha. Okay. So that, you can play that, you can download that from the internet easily. Craig has it up on his website and you can download that and play with it and look at it and you can freeze it and slow it down. And it's a very, very good way to study the process of this proof. So what they're arguing is, if the hat tiling were periodic, the, these two timings would be also naturally. Because as the edges shrink, the configurations don't change. What's next to what and so on remain the same. The clusters remain the same. And periodicity would be preserved. So if the hat were periodic, these timings would be too. But in that case, their lattices would be related by similarity. Now that takes some proving, but they do prove it. A similarity factor, so that it would just be a scale factor. But the scale factor has to take into account the differences in the, in their sizes. And in fact, it would require that the similarity be, have a scale factor of square root three over two. But in fact, that's impossible because all of these are based on hexagonal lattice and that becomes a contradiction. These are in commensurate with each other in that respect. So again, I'm skipping over the details here by say the least, but nevertheless, that is the argument. And so there we have a proof because this is an impossible thing and therefore these can't be periodic. Now what would Louie do with this? Louie would ask, how are these two proofs related? Does each imply the other? Is there a setting that includes them both? I like to think he would enjoy this birthday gift. And so I conclude just with my favorite picture of Louie and Therese, the adventure they made of life and their friendships and of intellectual activities and everything. Happy birthday to both of them. I wish they were here and thank you all. Thank you very much and best wishes to everybody.