 feel the elasticity in this thing. And this is really one of my main points for you, is that there is incredible beauty in the solutions to these simple partial differential equations, in my case the wave equation. So, here's a variation. What if instead of having congruent to one, mod two, you had them congruent to two, mod five, I misspoke. Well, that might look something like this. Yeah, that's two more than zero, that's two more than negative five. And it turns out that these are two terms in the Fourier series for a regular pentagram. So, if you write out the formula for the parameterization of the regular pentagram and compute the Fourier coefficients, you would find you need that much of this ingredient, that much of this ingredient, and that does a pretty good job of tracing the regular pentagram. And the fact that you have real coefficients here is what's causing that symmetry across the horizontal axis. Oops, I said vertical axis, sorry, horizontal axis. So, I took that trace and then I added on just some amount of this ingredient with the same symmetry. 17 is two more than five. So, this is going to preserve that five-fold, that special kind of five-fold symmetry that the other curve had. And I'm thinking of this as a path in a large design space, and it looks like this. Oh, you know, I find that hypnotic and beautiful. It has mirror symmetry exactly there, and then there, those are the two points where that coefficient is real. There's a Twitter bot by Michael Johansson's. Every four hours it makes a new curve like this. Please subscribe, it's really fun, it's a little oasis of, oh, that's beautiful on your Twitter feed. So, it's like way late to be talking about wallpaper, but let's talk about wallpaper. This symmetry work software, what do you do? You put in a photograph, you set function parameters, and you adjust them until you find an output pattern that you like. I took these sweet peas, which I grew, and put them through this process and turned them into this pattern. How does that work? Well, the basic ingredient is really nothing more complicated than the sine wave or that circular pattern. Yeah, I'm seeking functions that are invariant under two independent translations. The basic ingredient here is a plane wave, and I'm gonna say that it looks like this. Now, I'm thinking of waves like as the big waves in the ocean, not the waves crashing on the shore. I really don't like singularity, despite my example with the singular string. So, these are very placid, smooth waves. So, you could just think of these color bands as representing up and down of a very long, slow wave. This is the basic ingredient. So, that's like the sine arch. Now, I need to sort of get you into the land of complex variables here, and I'm sensitive to the fact that not everyone sees color in the same way. I am gonna use the words red, green, and blue because those are the things that are easy to program, but I hope you'll bear with me. What I'm seeing there is a red band, a green band, a blue band, a red grand, a green band, and a blue band. And these, I want you to think of them actually as the pure hues of a color wheel. Now, I've worked with students who see color differently, and many of them can tell the difference between the clockwise direction here and the counterclockwise direction. So, as long as you can tell that those are different, then you could get on board with this. I've taken those pure hues of an artist's color wheel. I've gradually graded them to white at the center and dark to the outer edges. And so, what formula do we know for something that goes round and around the circle like that? Well, that's the Euler formula that we studied for the circle. I've used the y variable here. See, this is constant across x's. So, I've just, that's really as e to the i y, just picking up those colors. Now, the fastest way I can explain the domain coloring algorithm is when I say complex number, you think, oh yeah, that's one of those colors. When it's the complex number zero, that's there. That's white. When it's the complex number infinity, then that's way out there that will be one of the dark colors. So, now, this is not the logical way to present this idea, but it's a very beautiful thing. Is that, now, the formula up there is that wave going this way. The second one is a wave going 120 degrees that way, and then there's a wave going 120 degrees that way. And I'm gonna add up those waves, and then I'm gonna use this coloring algorithm where I get a color and it turns into a, I get a complex number and it turns into a color. And oh, mercy, it looks like that. There, that is so rich. There's so much there. That's only three waves superimposed and they conspire together to make this beautiful wallpaper pattern. The formula gives a complex number at each point. That complex number gives a color. That is the result. I've created a wallpaper pattern just with three waves. Now, I wanna tell you a little bit about this one that the translational symmetries, and yeah, there's a little bit of a lie here. I have to rescale those waves I had, but the fundamental vectors are one, which is, see, there's zero to one. That's the real direction. And then this omega is a cube root of one, and it's just the vector pointing up 120 degrees that way. These are the translational symmetries. We mentioned that group idea that once you have some symmetries, you must have all the symmetries that devolve from them by composition. So I actually get a lattice of these translation vectors, meaning any integer times one plus any integer times omega. And then that is the ring of Eisenstein integers, which is the ring of integers in the complex number field to join the square root of minus three. So again, this situation where all things are connected, weird parts of mathematics crop up, I'll be mentioning the Eisenstein integers later. So if your goal was to construct all the functions that are periodic with respect to those two vectors, and you wanna have three-fold symmetry, then you'd use these building blocks for lattice symmetry. So I'm gonna introduce coordinates. X is a coordinate that sort of its lines, capital X equals constant are parallel to that omega vector. Little tricky to get those going. Y is just a variable that goes up this way, and those are called lattice coordinates. And you can write the complex variable z in this way using the lattice coordinates. So technical stuff here, but my formula for the waves is either the two pi i times nx plus my where n and m are two integers. So I've got those building blocks and I can superimpose these basic shapes. This is a Fourier series in two variables. Many people here could tell you interesting stories about the convergence of Fourier series in two variables. And just a note for later is that each one of these things, waves, knows how it should move into the future because it is part, it's the spatial part of a solution to the linear wave equation. So yeah, that's the wave equation that the acceleration should be the Laplacian. And the Laplacian is simply the sum of the two partial derivatives with respect to x and y here. Okay, so if you pick coefficients at random, you get something that has the right lattice symmetry, but it doesn't have a three-fold rotational symmetry. We already showed how to do that. You take a wave going this way, a wave going that way and a wave going that way. The formula for that is at the top of this page. And it's a little surprising that you take the nm wave and rotate it and you get the m minus n plus m wave. I was really surprised by that computation and pleased, but after all, the space of lattice things should be closed on the three-fold rotations if you're gonna have any hope of rotational symmetry. And this beautiful wave that I showed you before is the one zero wave. It's the one where n is one and m is zero. It's the bottom wave for my wallpaper functions. So there's your space to play in. You may choose these any coefficients and put them with these wave packets. That's one thing the software will do. You can use any kind of color wheel you wish. You can upload a color wheel. You can use predefined color wheels and you can get color patterings here that have the lattice symmetry and three-fold rotational symmetry. And again, just for emphasis, this is what the domain coloring algorithm is. The sum of wave formula gives you a complex number at each point, then that complex number gives you a color. You put that color in the domain of this function. Well, I realized long ago that I could do this, but I only accomplished it, I don't know, started being able to do this seven, eight years ago. Yeah, the world should be my color wheel. Why use these artificial artists' color wheels? Why not take that as my coloring is a complex plane? There's zero, there's one, there's I, there's two I. Why not? Combine it with waves, make a pattern like this. That's the essence of my work with wallpaper. But to lead on to something else, I wanna ask, well, how'd you create that reflection symmetry about the x-axis? Did you see that this has not only three-fold rotational symmetry about many points? But yeah, it's got, you can flip that one top to bottom. That's an additional symmetry in addition to the wave symmetry that I, or to the three-fold symmetry that I showed. Oh, here's a quick screenshot. This is currently what it looks like to experiment with symmetry works, is that you have all these ways to interact with the parameters. You don't have to be a mathematician. There's a place where you can grab a plane and move a complex number around that plane in order to determine a coefficient. And yeah, there's this project meeting that Jayadeva Treya is part of tomorrow at 315, where some students may wish to get involved with using this software to make wallpaper of their own, lots of variations on that idea. So yeah, when you try to make wallpaper, it's a matter of looking and then adjusting and then looking and adjusting and looking and adjusting. Scientific American said, can you work our logo into one of your patterns? And that was pretty hard. So I had to tilt the logo there until it sort of showed up here. This is, do you see the glide symmetry there where the surfer is gliding and going up? This shows that you can use an extremely minimalist color pattern and get an output pattern that's rather beautiful. There is a center of four-fold rotational symmetry. And then, so I was cutting beats and looked at my hand and said, I caught myself red-handed and had to take a picture of it and turned it into that, which is just, it's insane, that's a crazy pattern. But a thing that's kind of nice about it is that it has no other symmetries other than translational symmetry. So this is an example to show you can make a wild pattern without having any rotational symmetry, any mirror symmetry, nothing like that. So yeah, I want to just come, touch back on eigenvalues that here's the formula. If you take the Laplacian of this thing and divide out the function itself, this is wave by wave, you kick out this weird quadratic form, which is the quadratic form, the norm of the Eisenstein integers. And it only takes these values, you know, that's a list of the values that it takes on, complicated explanation by that. But potentially, if you were like, had a hexagonal wallpaper drum and could hear its vibrations, then this is the sequence of sounds you would hear. Now it's got all the same sounds a violin can make, but it's got some weird sounds in there. The one corresponding to the square root of three is weird. So here is a picture of one eigenspace where that's the eigenvalue seven. And like I'm thinking of the two three point here, that's two squared minus two times three plus three squared, that's seven. All of these dots work, the eigenvalue works out to be seven. That means all of these waves would vibrate at the same speed if they were vibrating according to the wave equation. There are 12 of these waves. I've colored them to show them locked into four triples. So like there's a blue triple here where lock those three together and you will create rotational symmetry. Something very beautiful can be seen if I think about additional symmetries because I was thinking about reflecting about the x-axis. Let's call that sigma x. It turns out that the action of sigma x swaps every blue for every red and every yellow for every green. So another moment of who makes this stuff up? That's to me exceedingly beautiful. So if you want to, oh, and yeah, the sigma y reflects in this way, all the reds swap with all the greens, the yellows and the blues. And then if you want to turn on six-fold rotational symmetry, then that is the last swapping you can do. This is a little Klein four group of things you can do. Product of any of two of these things is the other thing. And that really proves that there are five types of wallpaper with three-fold rotational symmetry. So if you wanted to have a recipe for wallpaper with that reflection symmetry that I showed, you would lock the waves together so that the coefficient of the NM wave is the same as the coefficient of the MN wave. So I've locked that together. I wanna give you just the shortest lightning introduction to classifying patterns by their symmetries. For some of the undergraduates, if you wanna get involved with this, you must learn to draw a fundamental cell. So when I look at this pattern, I see a center of three-fold rotational symmetry, three-fold symmetry, three-fold symmetry, three-fold symmetry. And those will create a minimal parallelogram that can stamp out the rest of the pattern. And it looks like that. See these yellow triangles, those are at centers of three-fold rotational symmetry. The blue lines are on mirror axes. Do you see that? The blue lines are on mirror axes. The green dotted lines may be difficult to see, but those are glide reflections that I bet you didn't even notice. If you flip this and slide it along that horizontal axis, there's glide symmetry here. So you learn to draw this fundamental cell and then you look up that one. This group, the group of all those things together is called P31M by the International Union crystallographers. Mostly, I love their notation, but this is appalling because it's subtly different from P3M1, which, okay, we won't talk about it. This is, but it's interesting, this giant group, this very infinite group, it can be generated by just two things. If you have a three-center with no mirror through it and then a mirror that doesn't touch that three-center, then that will generate all of these symmetries in this group. So the wallpaper surprise, I bet, for whom is it a surprise in this room? I hope someone, there's only 17 of these types. There's only 17 isomorphism classes of wallpaper groups. This is the diagram from my book of the wallpaper cells. So, and there's P31M, right? So you're keying out wallpaper, you draw this diagram and you see this. I love this fact so much. I participated in a chalk slam at Carleton College where they had wonderful chalkboards and were celebrating them, believe it or not. And so I drew all of the fundamental cells for the 17 groups. And then once you see these, you start to see them everywhere. So what movie is this from? While you're thinking, you draw the fundamental cell and see the rotational, two-fold rotational symmetry, that's diamonds, the parallel lines are mirrors, the glide is that dotted line. What movie is it? The shining, this is a PMG pattern from the shining. So I really want more than one person in this room to become the person who pauses the movie to look at the wallpaper. Okay? So please do that. I have a PDF available that sort of, you know, is a basic lesson in how to do that. So yeah, now we're time. Let me check the time. Cause we've come through wallpaper and we're in the end game here, color reversing symmetry. I've been talking to you about symmetries, which are patterns that leave, or transformations that leave a pattern the same. Well, I'm gonna show you some patterns where there's a transformation that inverts all the colors. In order to do it, I'm gonna use a color wheel like this. This is really kind of an ugly landscape, but it's made some surprisingly beautiful patterns for me. I've taken its photographic inverse here and replaced it upside down. I'll show you a functional equation in a minute that shows you why I did that, but that turns into this pattern here somewhere. Yeah, there's the functional equation. And in another slide, I don't have my mirror in the right place. It's that diagonal. I'm gonna call that the diagonal mirror. And when you apply that to the argument of the function and then evaluate the function, you get the negative of what you would have had without evaluating the function. The algebra of color reversing symmetries is stunning. I recommend it highly. Yeah, there's my mirror there. And to classify them, you say, yeah, the actual symmetry group, the transformations that leave the pattern the same, that's P4. But if you were to declare that colors and their negatives are the same, or in other words, if you were to think about the symmetries of the absolute value of that function, then you'd get the group called P4G. You might have to look that one up and draw its fundamental cell. So I'm gonna call that P4G over P4. How many of those types are there? Well, here's another one. And I didn't really tell you the story. This is a stained glass window in my house and I've made so much out of it. But I love this pattern because it looks like these are in the foreground and those are in the background. So the algebra says that the symmetry group is normal in the color group. And so all you have to do to answer my question is quickly compute how many ways can you think of for one wallpaper group to be normal in another one? No, it takes a while to work that out. In this one, we've got this color reversing half turn about that point. You turn that one, turns into that one. The actual group symmetry group is called P3. That's just plain old three-fold rotational symmetry. If you allow that half turn as a symmetry, then you get a larger group P6. This is type P6 over P3. There are actually 46 color reversing types. So to make yourself a sampler with one of every type, that's quite a chore. This fact was first published in the Journal of Textiles, Manchester in 1936. Because people were interested in patterns they could make with the Jacquard loom. And there's a project available if you wanna work on color reversing symmetry. Just the quickest look at three color symmetry. This is a beautiful steak dinner I had in Perth, Scotland. I did violence to it by rotating it and making these weird colors. And this pattern, is that weird? Don't I get to do weird things? You know, I love that pattern, but it's too weird. So yeah, the symmetry group is normal in the color group with index three. How many, and here's an appetizer of hummus that's turned into a four-fold symmetric thing. Get these mysterious hummus birds here. And here the functional equation is like this. Okay, well let's rock it on. Symmetry variations, the end here. I took this picture of a peach, made a color reversing color wheel of it. This is actually basically the cover of my book. Let me show you a close-up of the left-hand side here. That that, see the blue picking up? That's the negative of the peach. So we have blues by sixes and peaches by threes. Well, all I did was gradually take that photograph and just as I went across, I gradually turned it upside down. I told the computer to turn that upside down so that by the time I got over here, I had peaches by sixes and blues by threes. This is a morphing freeze, where a pattern turns in to its exact opposite. Yeah, hyperbolic wallpaper. It's a whole another story and I can't begin to tell you, but that's also out of that same photograph of a peach. And I wanna thank Jeffrey Hofstein, who is here for helping me make my first hyperbolic wallpaper, but we need to move on. I've been working on a project with Jeff Weeks about polyhedral symmetry using harmonics. This is, it's not like we're saying anything new or revolutionary about spherical harmonics. It's just, we found a new way to connect something that he was doing a decade ago connected to cosmology with something that I was doing with mathematical art. So, and this is one of my favorite works to date. It's a bookshelf with six spheres on it. Of course, these are not real objects. These are computed spherical textures, computed in order to have polyhedral symmetry. In Photoshop, I know how to shine lights on them, create surfaces to reflect off, so you see the reflections of these balls as if they're sitting in sort of a mirror box like this. Let me tell you a story about just one of these balls. It's gonna be the middle one up here. You start by taking an image with a spherical camera. Henry Segerman turned me on to the camera that takes a photograph of the entire sphere. And this is the Lakewood Chapel in Minneapolis, Minnesota. Anybody know Lakewood Chapel? It's, that's, you know, looking at the altar. And if you look directly behind you, you'd look out that door. But in this photograph, it's taken it spread out like this. Well, then I used this function with cube symmetry and cube mirror symmetry in order to create this. And I wanted to show you a closeup of how, you know, it's just darling. You can see the mosaic work here coming around like this. And that figure is actually the tripod that the camera was on. So it becomes a feature of the design. And okay, my almost last topic is about Fibonacci spirals that I've been doing recently. If you got the right wallpaper, it's gotta be oriented the right way. You can use the complex exponential map to wind that around the plane and get a Fibonacci spiral like this. It's the complex exponential. I owe a lot to John Edmark. How many people have watched John Edmark's viral videos called blue?