 Professor Frank Ferris, who teaches at Santa Clara University. I've known Frank longer than almost any other mathematician. I met him when I was a freshman in college. He was considerably older at that time. And he's near, we were just down the road for me in Stanford. So he's done many things in his life, but in the past several years, he's done some very beautiful work in symmetry, using mathematics to generate symmetry. And he published a book, which is called Creating Symmetry, the Art of Mathematical Wall, Paper Patterns, which has received wonderful reviews and is a terrific book. And I'm very pleased to introduce Frank Ferris. Thanks, Ray. Can you hear me all right? When I think about the problems ahead for our planet and for the United States of America, I'm sometimes overcome with grief and fear. But I take refuge in the beauty and constancy of mathematics. And when I look out at this audience and see people who are going to spend the next three weeks learning something new, gathering their energies, to take those energies out to their communities, then I'm filled with hope. So let's all, at all levels, embrace teaching as a radical activity for social change. Thank you for being here and thank you for coming to my talk. I wanted to say that I will be here through Thursday night. I would love to talk to you individually or in small groups. If there's anything that I could do to help you create a mathematical visualization, make something beautiful for a talk you're going to give, or something wonderful to take home and share with your students, I would be happy to help. Before I begin the mathematical part of my talk, and there really is quite a lot of mathematics in this talk, I want to give you an overview of my mathematical process. I make patterns. Oh, we're going to have these lights off, these first row of the house lights. I would much rather you look at the screen than at me. Thank you. So my overview is that I make patterns out of wave functions. And this is a meeting about harmonic analysis. So the waves fit right in. And the way I say it to a lay audience is that I start with a photograph like this, this beautiful peach on a rock. That's my leg and the stream that I climbed up to eat this peach for lunch. And I use mathematical waveforms to turn it into a pattern. For mathematicians, this is a little bit backward. For mathematicians, I would rather say that this is the domain of the function. The pattern is coloring the domain of a function. And that function takes values in that photograph. How do you find functions like this that make patterns? Well, the key is harmonic analysis. But first, it's kind of fun to look at a real close-up of a pattern like this, where you can see the little nub of the peach and a little piece of its stem have become beautiful decorative features of this pattern, including the texture of the denim of my jeans and even the little laces of my shoes. They become an important part of this pattern. Now, yes, it takes certain software to compute these patterns. These are made in C++. But there's a software package that is very easy to use. It's called Symmetry Works. You could play too. So one hope is that some of you will go home and become makers of wallpaper patterns, an activity that I enjoy very much. This is public domain software that was written two summers ago by students at Bowdoin College based on a user interface written by a Santa Clara student, and I have a student at home now working to improve it further. So I want to quickly go over the three major categories of playing patterns, but I need some vocabulary. I need the word symmetry. For me, a symmetry is a transformation of the plane so that when you transform the plane, the pattern remains unchanged. That is a symmetry of this pattern. This pattern is called a rosette, because every symmetry of this pattern leaves one point unchanged, the point right at the center. And I made this one out of a photograph of carrots and bell peppers on a cutting board. Cut them up, thought they were beautiful, took a picture of them. I'm going to try to soft pedal the algebra side. This is an analysis group, but I do have to say that the symmetries of any pattern form a group. Now, if you don't know what a group is, this is the first best example for you to think about. A group is a set that is closed under a certain operation. Here, the set is these transformations that leave the pattern the same. What is the operation? It's composition of those. So this pattern stays the same when you rotate it by 60 degrees. What if you rotate it by 60 degrees again? Well, it stayed the same the first time. It stays the same the second time. So under the composition of those two rotations, it stayed the same. It turns out that there are exactly six things you can do that leave this pattern the same, a group that many people know as C6. So I'm not going to say too much about groups, but I wanted to just give an example of the correct use of that word. The second category of plane patterns is freezes. A freeze is where all of the symmetry, let's see, the translational symmetries all proceed along one axis. So here, left and right. And yes, I'm inviting you to use your wonderful visual cortex to realize that beyond here, there is another cycle of this pattern, and another, and another, and another. And likewise, back this way, too. So this was made from a photograph of my living room before a party with a bowl of cherries and strawberries on the table. This and then wallpaper is sort of my main gig wallpaper. A wallpaper pattern is one with translational symmetry in two linearly independent directions. So I can tell you a little bit of my origin story as a mathematical artist by telling you about a teaching experience I had in the mid-90s. It was a grating definition of a pattern. I was teaching geometry, and there was a little section on freeze symmetries. And this book said, a freeze pattern is a set of points that is invariant under blah, blah, under the translations under a certain group. And their illustration was something like this. Now, this is a good illustration in that you can see the intent to have translational symmetry of one unit, two units, three units, and so on. There's an additional interesting symmetry that I wonder if it's familiar to you. You can flip this one that does not preserve the pattern, but if you flip and slide half a unit, the pattern is preserved. This is called glide symmetry. So I look at this, and I say, but wait, what about the beautiful artistic freezes that have been created by artists for many centuries? I think that patterns should be functions. So it was that teaching moment that set me, I think 25 years ago, onto this long exploration to say, no, you should make your freeze like this. And that is a function on a strip of the plane whose values are in that photograph of an Indian paintbrush from the Sierra Nevada. Another grading definition of pattern, as I learned more about it, I read some of the great books about patterns, and this definition said, a pattern is obtained by taking a motif and repeating copies of it without overlap to fill the plane. If you want a pattern in the plane or a rosette, you could take a slice of pie and repeat it around. So I said, okay, fine. If you want to make that kind of pattern, you can go stamp, stamp, stamp, stamp, stamp, fine. That's your pattern using the strategy of a motif. But I think a pattern should look like this. That's my idea of a rosette pattern with five-fold rotational symmetry. And sure, it's technically true that you can cut a slice of the pie, really any slice of that pie that's 72 degrees at the point will, you can make the whole pattern by repeating that slice of pie. So that's where I come from in this idea about using functions is that for me, they just offer so much more artistic potential. I'm gonna say a lot of things today that are gonna go very quickly. You can read the details in my book. The title is Creating Symmetry, the Artful Mathematics of Wallpaper Patterns. Came out from Princeton University Press in 2015. I've been delighted by the reception of this book. So that's available to you. So here's my list of topics that we're gonna go over. I'm going to tell you about the vibrating string and circle as a first way to get involved with the theory of waves. Then we'll talk about wallpaper waves, which requires me to tell you about something I call the domain coloring algorithm. And then we're gonna talk about color symmetry in wallpaper. And then I can't resist showing you some of my work since the book came out with things that are beyond plain patterns. Quite a lot has happened. And really there are a lot of things for you to engage with as well. As I talk about these topics, I'll be mentioning a few to things that some of the undergraduates who may be looking for short term research projects may wish to pick up on. So yeah, the vibrating string. Imagine a string like a violin string that is tied down at both ends. It's free to vibrate up and down. For technical reasons, I'm going to assume that it only vibrates up and down, not side to side. This is most realistic if the vertical dimension that's shown here is really quite smaller than is shown here. So that's the formula for it. This is the simple arch of a sine curve, just like that. And here's my first space of functions. I hope the mathematical notation is not too intimidating. This says the set of all functions that live on the interval from zero to pi with real values to indicate how high up or how high down with the property that they are tied down at the ends. That's a space of functions in which I may look for my string patterns. And the thing about this string is if you, it's been hanging there at rest for quite a while, but I can release it from rest and it will naturally vibrate like this. Up and down, up and down. Now, I have exaggerated the vertical displacement. The mathematical model is not really realistic for such a large displacement. But what do you think? I think this is one of the platonically beautiful things that the human mind just sort of adores, the motion of a simple sine wave going up and down. The thing, in order to make motions like this, I should try to solve what's called the linear wave equation. And if partial differential equations are too much, well, this means the vertical acceleration of the position of the string. And this represents the concavity of the string. So if you think about it, if the string is concave down, it should be accelerating down. If the string is concave up, it should be accelerating up. So that's a very simple interpretation of the linear wave equation. And here is my solution. I have taken my simple spatial sine function and multiplied it by the cosine of t. And people who know the technicalities by this will see that for this and all of the motions I'm going to show you, I'm beginning a motion from rest. And that's why I use the cosine here instead of the sine. But let's just pause. I bet you can differentiate this twice with respect to x. Sine goes to cosine, cosine goes to minus sine. That kicked out of minus sine. Start here, cosine goes to minus sine, minus sine goes to minus cosine. That kicked out of minus sine too. So this really does, both sides of that wave equation are true. So then I'm going to ask you, what other functions can you think of that would make this happen? So yeah, there they are. You could scale these by integers. And I don't have it on my slide, but n should be an integer one, two, three. Why is that so? Because I required the string to be tied down at both ends. Here are the first three of them. You can see the placid red one and then the somewhat faster blue one going twice as fast and the green one going three times as fast. And then there's this magic moment there where they all go back to their starting positions. It turns out this is unusual in the theory of the vibrations of things to have them periodic, that there's a moment where they all return to their starting point. But this is a very special situation. Sometimes I think that this is why the counting numbers exist. It's not because we have five fingers, but it's because there are modes of vibration of a vibrating string that are in correspondence with the numbers one, two, three, four, five, and so on. That's why we have counting numbers. And I think the next thing that's supposed to happen is to play that little icon so we've got somebody on the sound. That sequence is the basis of at least all western music and really indeed quite a lot of all human music. So I think that this is an iconic thing. This is part of our intellectual heritage, the existence of this sequence of string vibrations. In order to generally solve the wave equation, you wanna superimpose these with coefficients. And that's the thing about linearity. It's hard to get across the emotional impact of linearity. It's just so beautiful that once you have some solutions, you can superimpose them like this, meaning add up a sum with coefficients in front. And so I did some adding, oh yeah, no, I wanted vocabulary. These are called eigenfunctions of the Laplacian. In this context, the Laplacian is just a fancy name for the second derivative. And what does eigenfunction mean? It means you apply that operator, meaning differentiate twice, and you get a multiple of the function, German eigen to the self. These are functions that go to themselves with a little multiple here. And the eigenvalues here are the negatives of the squares of these counting numbers, one, two, three. When we think about frequencies, we're gonna think about the square roots of the negatives of these eigenvalues. So when we think about frequencies, we're thinking about one, two, three, and so on. So I just made a little pattern here. In making art, it turns out that it's best to include smaller amounts of higher frequencies that makes, to my taste, prettier patterns. So I made one amount of the first building block, half amount of the second, third amount of the third, and I made this shape over here. So, and you could think of this as the starting point in a linear space of functions. That's my function space. And say, here's where I wanted to start. Well, each of these waves knows individually how to move forward in time. And when they move forward in time, they look like this. And again, this has this graceful dancing flow. It's, this equation is somehow right because it is showing us something that we interpret as, oh, that's graceful, that's a beautiful dancing motion. So not to spend too much time on this base case, but it turns out that you can make pretty much any starting function from base waves. Experts will correct me. But for instance, I continue, do you see that this is a continuation of that pattern where I had one, a half, a third, a quarter of sine 4x, and so on? It turns out that if you sum those all, it really does add up to that linear function, which is tall at the left and zero at the right. And that, oh, when I take 10 terms of this Fourier series, it makes a pattern like this. And right there, it's trying to make that linear function sloping down. Right there, it's trying to make that slope down linear function. There's something weird at the left-hand endpoint. Think about the slopes of these at zero. The slope of this, each one of these building blocks is one at zero. So you're adding up a lot of very high slopes there. Here it is with 50 terms, and you see something weird going on. Yeah, and experts will be able to tell you about the propagation of singularities observable here. Notice that I'm asking these simple sine waves to do something very unfair. I'm asking them to jump from zero up to pi over two, actually, right in an instant there. It can't be done. But these waves, they do so well to try and do it. Now, you'll notice this little overshoot when they jump here, they're overshooting. I know that one of the projects proposed for the undergraduate research projects is to study that overshoot, it's called the Gibbs Phenomenon. And then just to take delight in the connectedness of all mathematics, if you plug in right at the middle, plug in pi over two there, it turns into the Leibniz series for the number pi over four. So, who makes this stuff up kind of moments, the connectedness of all things? So the summary of the string is that, and we're gonna see this in many situations, is that you discover some shapes, which are eigenfunctions of the Laplacian. You create your desired starting point from those shapes, and then each individual shape knows how to move forward in time. Let's take a leap to the circle. So, I hope people might recognize these as the parametric equations for the circle, cosine t, sine t. As t progresses, this starts at one there, and just goes round and round and round the circle. Well, I wanna take you into the world of complex numbers by saying, you know, complex numbers are just a repackaging of Cartesian coordinates. You just have this other little slot i there, and that's the real part, that's the imaginary part, that's the x coordinate, that's the y coordinate, and we're gonna do algebra on these with the requirement that i squared is negative one. And then you're gonna take another leap by saying, you know what, there's a shorthand for this, which is e to the i t. And this is the beautiful Euler formula that I think many of you know all too well. Anyone could write quite a long article about how you might think about going from there to there. It's the easiest, non-trivial example of a map from a Lie algebra to a Lie group. So, anyway, I'm gonna be using that shorthand. Another thing about it is that that obeys the usual rules for exponentiation that you can exponentiate a sum by multiplying the two individual exponentiations. And that also is an eigenfunction of Laplacian. So I'm gonna ask you to believe that you can do a differentiation rule where you pull down that i, and if you pull down that i twice, the eigenvalue is minus one there. So this too is an eigenfunction of the Laplacian. If I think of Laplacian as being simply second derivative with respect to t here. Well, and then I'm gonna do this trick with periodic time rescalings. That's what I did when I went from sine x to sine two x to sine three x. Here, I'm going to scale these by going once around, twice around, three times around. And yes, negative one times around, negative two times around, that means going around backward. So this now is my space of functions to play in. And my discrete question mark here is this really the same as all periodic functions? Can any periodic function from the plane, from the real line into the complex plane, can it be written as one of these exponential sums? Well, go ask a harmonic analyst to reveal to you the amazing subtlety of that question. There are, it depends on what you mean by is, right? So here's another part of my origin story is that long ago in the 90s, I was using some simple software called DRIVE to show parametric equations to my students. And I wrote out what amounted, if you put it in complex notation, it's this. And I thought I was just making up a random example. And Jerry Alexandersen, who's been a lifelong friend and mentor and who has recently retired after 60 years at Santa Clara, said, well, Frank, what did you do to create the symmetry? They said, I don't know. When you rotate from there to there by 72 degrees, the pattern falls into coincidence with itself. Indeed, the trace of this curve does have five-fold rotational symmetry. Here's the functional equation for that symmetry. The left-hand side says, let's advance time by a fifth of a full period, two pi over five. When you do that, you should get back the original thing you get, but multiplied by this rotating factor. So that multiplication factor is a rotation through 72 degrees. So you start at the red arrow, you advance time, and that is the same as rotating that point to the second red arrow there. How do you solve that functional equation? Well, this was my first article in Mathematics Magazine, and which led me to actually later be editor of that magazine for many years. And so you can read the story. I'll tell you the punchline though, which is, let's see, there's the functional equation we're trying to satisfy. These are the kinds of functions we can use, and this is an example of something that works. Well, when you think about kicking out a factor of two pi over five in the context of putting that six there, well, five of those are gonna give one full rotation which you throw away, and there's one left over. So throw away by fives and have one left over, that means that your coefficient should be nothing unless n is congruent to one mod five. And what about this negative 14? That's one more than a multiple of negative 15. Oh, that does fit the recipe. And then I just have to tell you that, you know, once you've got waves like this, each one knows how to move forward in time, and I'm gonna leave that one in place. That is the circle. I'm gonna think of the other terms as deformations of the circle, and I'm gonna let them rotate forward in time. Oops, I need that word. This is called a recipe. This is a prescription about coefficients in a series that will guarantee a kind of symmetry. That's what I'm gonna mean by a recipe. So these waves know how to move forward in time, and it looks like, oh, that beautiful bounce. Now, I'm totally violating anything about the wave equation, but nonetheless, there's something platonically beautiful about this in that moment where it turns around, you can just,