 And now I'd like to bring up our introducer for tonight from Sarah Lawrence College. I'm delighted that he agreed to come and introduce Frank. And so I give you Dan King. Hey, everybody. Good evening. You're in for a real treat. I, too, heard the earlier talk. And I would describe it as stupendous and stunning, stupendous mathematical ideas generating some stunning art. So I think you're going to really enjoy this. Our special speaker is Frank Ferris. He is professor of mathematics and computer science at Santa Clara University. That's in California. He's been teaching there since 1984. He did his undergraduate work at Pomona College and got his PhD at MIT. And he is a triple threat. He has won awards for not only his writing, but also for his art and his teaching. He won the MAA, that's the Math Association for America's Trevor Award for Mathematical Exposition. You'll definitely want to check out his book, Creating, which is right over here, Creating Symmetry, The Artful Mathematics of Wallpaper Pattern. But he also won a recent award for his art at the most recent 2018 joint mathematics meeting. He won specifically the best photo, print, or painting at that conference. And perhaps most impressively is his collection of teaching awards. He won the teaching award at his institution, Santa Clara University. But just hot off the press, he also has won the teaching award for the golden section of the Mathematical Association of America, covering all of Northern California, Nevada, and Hawaii. He's certainly a great teacher. And I know he's a great speaker. So help me in welcoming Frank Farris. Thanks so much, Dan. Thanks to all of you for coming. For mathematicians, there are a few things as pleasurable as having a roomful of people who actually want to be there to hear what we have to say. Thank you for coming. And thanks to this wonderful museum and its dynamic director, Cindy Lawrence, for the invitation and for making this possible. I've been on a wonderful journey for about 20 years of becoming a mathematical artist. It's not something that I set out to do. I feel as if it's something that happened to me, thanks to the wonderful qualities of mathematics. So let me tell you about this. I want to start by telling you about symmetry. So let's ask, what does symmetry mean? Well, years ago, about 20 years ago, I asked one of our art historians at Santa Clara University, how do art historians understand the concept of symmetry? And she said, it's like the two sides of a face when they're the same on both sides or the two wings of a butterfly. Or like the two halves of the front of our mission church. This is the campus that I left behind in order to come here. It's a beautiful, beautiful campus. And this is typical mission style architecture. And you can see that on the facade, the both halves are the same. Someone might say the tower breaks the symmetry because there are not towers on the same side. This concept of symmetry comes from nature. Many things in nature are symmetrical. Like for instance, this beautiful Mariposa lily, it does, you can see that it has two sides the same. It's the same on both sides. So that fits the art historian's view. One thing that differs from this art historian's view that symmetry always means two things the same on the same side is some of the decorations that we can find on the inside of the mission church. For instance, there's this little decorative rosette, which yes, it's the same side to side, but it also consists of four parts the same. I would call this a rosette that has four fold rotational symmetry. Looking up at the ceiling of the mission, there's this beautiful freeze pattern that is rather simple. It's a pattern that repeats indefinitely as many times to the right or the left as you wish. So this is many parts the same in a repeating ad infinitum way. Also on the ceiling, there are some beautiful wallpapers. Again, somewhat primitive style, but this pattern is one that repeats side to side indefinitely as well as up and down indefinitely. So in a way this fits the idea of the art historian of sameness of parts. Yeah, this involves somehow something made of parts the same. And so that's our generalization of the concept of symmetry, but mathematicians use a little bit different vocabulary. And it's this vocabulary that really got me into this whole thing to begin with. It's thinking of symmetry as being about invariance. So if I look back at this lily and think about the idea of invariance, I think about it as, yeah, if you were to flip that photo over, that act of flipping would not change the shape of this lily. Furthermore, if you were to rotate that photograph by 120 degrees and focus just on the lily alone, that would not change the pattern. So that shape is invariant under rotations by 120 degrees as well as this flip. Now by exploiting the concept of invariance, I found ways to make my own patterns that echo these same concepts of symmetry. For instance, here is a thing so that when you rotate it by 72 degrees, the pattern falls into coincidence with itself. Now if you truly wished, you could think of that as a thing that is the same in five parts. But I would rather you think about that as a thing that when you rotate it one fifth of the way around, the pattern falls into coincidence with itself. Similarly, I have ways to make patterns that repeat side to side. This is a pattern that repeats side to side and it has another couple of kinds of symmetries. When you reflect about a vertical axis, the entire pattern stays the same. And I guess if you flip it top to bottom, it doesn't match, but then if you shift it over, it does. I've also found ways to make a pattern like this that repeats from side to side as well as in a different direction that seems to be sort of an offset upward direction. But this does repeat ad infinitum, if you have the imagination to think of this as being a swatch of an infinite piece. It repeats forever in two independent directions. And that's the defining property of what I wanna call wallpaper. So I'm gonna call this a wallpaper pattern. Notice that it doesn't need to be pasted on a wall. It could be a carpet on the floor. It could be on your computer screen or it really could just be in your imagination. But by the term wallpaper, mathematicians refer to a pattern that changes or that is invariant under shifts side to side and in one other independent direction. With these ideas in mind, I can tell you my origin story, how I got into this. It all happened because I was teaching out of a textbook that had a definition of the word pattern in it and I just hated the definition. So maybe you will be so lucky as to find something that makes you angry enough to start working on something for 20 years. So it was a wonderful beginning. But this book said a freeze pattern, so freeze is the word for the things that continue side to side. A freeze pattern is a set of points that is invariant under these side to side motions and blah, blah, blah, other things. Set of points, set of points and the pictures in that book all look pretty much like this. So that does get the idea across of repetition side to side. You might also see that when you flip this one top to bottom, the pattern does not match until you shift it over half of a cycle. So yes, this does display the concept of invariance and so I would say that that image has symmetry but that's just not the kind of image that I wanna show my students. I wanna show my, the set of points, how is a pattern? The set of points, I was, we're all aware of decorative patterns that we've seen out there created by actual artists, not by mathematicians. I wanted to make patterns like this and so I invented a way and did it. So I like this as a demonstration of a pattern that repeats side to side and has that additional feature when you flip it top to bottom and move it half a cycle either right or left, the pattern falls into coincidence with itself. So I got very interested in making patterns that were beautiful and that were not just about having parts the same that went beyond the idea of a pattern as a simple set of points. Just for one more detail about my origin story, if you read certain texts about symmetry, they will talk about a pattern as consisting of repetitions of a motif, a motif being a little piece of a pattern and then to make the pattern you repeat, you go piece, piece, piece, piece, piece. Well, if that's what you want in a pattern, you can make one of those, fine. Stamp, stamp, stamp, stamp, stamp. If that's what you want out of a pattern, if you're gonna focus on the concept of motif, then you can make that. But I wanted to make this. Now, I know that some of you are closer to a screen than to the front here. If you can see, this is a rather humorous self-portrait that I've used a photograph of myself and there are five copies of me, there are actually 20 copies of me circulating around there. But the thing that I'd like you to notice about this is that you could cut it up into five pieces all the same. It's still true that this has a property of being five pieces the same, but that's not how it looks. The way it looks is like an object that is invariant under a 72-degree rotation. That's how it looks. And it's seamless as it does this. Now, my background is in continuous mathematics as opposed to discrete mathematics, continuous mathematics, the mathematics of growth and smooth change. Much of mathematical art has been devoted to discrete mathematics, which produces images that look a little bit more on the left. So this has been my long journey is to apply my background in continuous mathematics to make patterns that look more like invariance than they look like many parts the same. If we focus less on individual pieces and more on invariance, then we might be able to make patterns from waves. So in my undergraduate education, one of the delightful experiences was Fourier theory, learning about Fourier series and how everything is constructed from waves. And I've been very happy to find a way to apply that to mathematical art. So I found a way to start with a photograph. This is Fireweed in Belfast, where I was lucky to go last summer. I take a process of mathematical waves and turn that into a wallpaper pattern like this. You can see that it's wallpaper because it repeats the same in two independent directions. Oh yeah, and it's like a kaleidoscope, but without mirrors. So if you've looked at a kaleidoscope, you can see that material from the world is turned into a pattern. But the way a kaleidoscope works physically is to show you mirrors that meet at an angle and that will always produce patterns that have these square edges in them. And as a person who comes from the land of smooth, I do not like those edges in things. And so I'm more delighted by a pattern like this. And it's really fun to look closely at these. I've really enjoyed printing some of my work on large aluminum panels where you can see the detail. So here I've enlarged, I hope you can see, the stamens and pistols of the Fireweed appear as really interesting lacy detail of the pattern. And you can also see up close these sort of angels in spaceships kind of features of this pattern. So it's really fun to look up closely at this. And I have to tell you, you don't have to know a lot of mathematics to make patterns like this. In fact, I've got this free open-source symmetry work software that would allow you to play, too, to take your own photograph and turn it into wallpaper. And I have a handout about that if you're interested in doing that. This was public domain software written by students at Bowdoin College under a grant from Bowdoin two summers ago. So there is a handout and you can also find this information in many places. So if you wanna know the mathematical details of my story tonight, I invite you to read my book. I'm very flattered to see a display of books here. I was delighted by the process of publishing this through Princeton University Press. Got a lot of help from them. And this is a very mathematical book. It's really got all the mathematical details. So if there's something tonight where I go more quickly, just to fit things into the time that we have, know that all the details are really explained in this book. So that was the long introduction. And here's what we're going to do this evening. I wanna talk to you about how to look at a pattern. I wanna help you see what to see in a pattern, okay? Then with that in mind, I wanna help you think about how to classify patterns. There is a mathematical classification scheme for patterns that is very strong and I think beautiful. Anthropologist Dorothy Washburn even recommends that this classification scheme should be used by museums in order to classify, for instance, their textiles or their pots with patterns on them. So that's my learning goal for you for tonight is that you become more able to classify patterns according to this mathematical scheme. So yeah, we're gonna ask you to try it. We have some handouts with patterns where I'm gonna see if you can classify those patterns. Then after that break, I'll tell you the basics of what I mean by the wave functions that make these wallpaper patterns. And then finally, since the book came out, I've been working in a lot of other areas of mathematical art. So beyond plain patterns, I'll tell you some of my adventures beyond wallpaper. There's a theme that just is going to keep coming at us again and again. And this is the theme of freedom and constraint. So every artist wants to think that they have artistic freedom to do anything they wish, but in every medium, there are artistic constraints, things that you may do, things that you must do, things that you may not do. And I wanna show that in this art form, there is also a very interesting interplay between freedom and constraint as we try to make wallpaper patterns. So, here's our first pattern to look at and to say, what do we see in this pattern? The first thing I'd like to call to your attention is the fact that your mind will inevitably identify this as a pattern. You will look at that and say, look, it's a pattern. Now, what is responsible for this? I assert that your mind knows how to tell you what happens beyond the right-hand edge of that pattern. How does your mind know how to do that? I believe it's because your mind can pick up a piece of the pattern and move it and see, yes, that's the same. So I'm inferring that as I go beyond that frame, it will continue in the same way. I want you to see this as a representation of an infinite pattern. How do our minds enable us to do that? Where is it in our human imagination that we just know that, oh yeah, that represents a swatch of something infinite, and that's where our inner mathematicians take delight in that. I worked with an Islamic student who said, in our culture we make these patterns because they remind us of the infinitude and immanence of God, God being everywhere and extending infinitely. So I find that moving. How did your visual cortex evolve so that you could pick up a piece of this and move it and recognize it as being the same as something else? Well, I don't know how that evolved, but I know what mathematicians call that. We call it translational symmetry. So I've picked up one piece of this pattern and moved it down and to the right, and I see that the pattern falls into coincidence with itself. And if I really were able to fit my infinite projection here in this small room, the infinite pattern would match as well. So, and there's a little artifact of PowerPoint in here where it doesn't quite match. I think that's almost good. If I did it in Photoshop, I could make that match absolutely seamlessly. So, and you might see this as translation to the right, and then translation sort of down a little bit, or you could see it as going just translation down to the right. So there are several ways to see that as a particular translational symmetry of this pattern. A symmetry being something that I am able to do to the pattern that causes it to fall into coincidence with itself. So this just shows one of the many translational symmetries enjoyed by this pattern. I can't credit the author, but I remember the first time I read someone saying, this pattern enjoys many symmetries. I thought that was a very nice way to say it. So you wanna call out where I'm gonna go next with the symmetries of this pattern? They are already in order, but six did I hear? We'll get there. I'm gonna do mirrors first. So your mind has the ability to look on both sides of that red line and say, you know that pattern is exactly the same on both sides. If I were to do the operation of flipping that pattern over, the pattern would fall into coincidence with itself. The pattern is invariant under that flip. I'm gonna call that mirror symmetry. So this is one of many mirror symmetries this pattern enjoys. And that's how I'm using this word. A symmetry of a pattern is a transformation of the plane so that when you do that transformation, the pattern is invariant. I wanna call that line a mirror axis. There's also another kind of symmetry here that is very similar to a mirror reflection, but it's called a glide symmetry. What does your mind tell you about reflecting across that red line? The pattern does not match on both sides of the red line unless you slide it down. If you flip and slide, the pattern falls into coincidence with itself. This is called a glide reflection. There are many glide axes here. I've drawn your attention to one of them. And yeah, when I think about glide symmetries, I might think about footprints with an alternating rhythm like that. So here you might guide your eye to say, here, here, here, same visual rhythm as footprints. So that's called a glide axis. Now later, keep in mind, I'm gonna be handing out some patterns and asking you to identify, do I see mirror symmetry there? Do I see any glide reflections? There are also rotational symmetries. So here, I've pinned that upper right corner at the center of that weird flower and allowed the transparency to sort of slide down by an angle of exactly 60 degrees. Why 60 degrees? Because 360 divided by six, 60 degrees. And the pattern apparently falls into coincidence with itself. The pattern is invariant under this rotation of through 60 degrees about that point. So I wanna use some vocabulary about rotational symmetries. And these are some words that you will definitely need later as I ask you to try this yourself. When I point it to that point and call it a six center, that means that when I rotate about that point by one sixth of a full rotation, the pattern is unchanged. So in sense, 360 over six is 60 degrees. That's why it's a 60 degree rotation is a sixth center. Who sees another kind of rotational symmetry next in order is three, thank you. So there's my three center just like that. So that's gonna be 120 degree rotation. And you can do that three times and that will pick up a full revolution. So in looking at patterns, you might keep your eyes open for six centers or three centers or two centers. Or four centers. Well, now, but in this pattern, do you see any four centers? I do not, okay? So this pattern seems to have six centers, three centers and two centers. And it makes an interesting question of, are you free to select these symmetries independently? Suppose you want it to be exceptionally creative and say, I really like the six-fold rotational symmetry, but I don't wanna get mixed up with that three-fold or two-fold. Would it be possible to make a pattern that has only six-fold rotational symmetry but none of the three-fold or two-fold rotational symmetry? Open question, but of course, that connects to our theme of freedom and constraint. And then to tell you a little bit about the artistic side, a few weeks ago I visited a beautiful garden in our peninsula in the Bay Area called Phyloli and took a picture of this hellebore. I saw some hellebores blooming by the Natural History Museum on Saturday, that beautiful day. It's a flower with really not very much color variation, but I really love the color gradient in it and the way the purple gets in there. So I use that photograph to make this pattern. As you think about becoming a person who knows how to classify patterns, I want you to be aware that there are three major categories of plain patterns in Euclidean geometry. The first is the pattern called a rosette where all the symmetries, those are the things you can do that leave the pattern unchanged. All of those symmetries fix one particular point and a typical rosette looks like this. This was the poster for this talk, so presumably you've seen it. So this, all of the symmetries, let's see, I could name for you in this case, all of the symmetries of this pattern. Rotate by once by 72, twice, three times, four times, five times your back and it's if you had not rotated at all. So there are five symmetries of this pattern and that's all there are and that's a rosette. Freezes are patterns with translational symmetry along one axis. And freezes have been a little bit neglected in my work. They're nice, but they're not wallpaper. So I'll show you this one. This has that pattern of yes, it translates back and forth and if you flip it, it doesn't match but then if you slide it, it does. And then to show you about the photographic source for this pattern, I'll show you a paintbrush grown in the Sierra Nevada. And then there's the star of tonight's show which is wallpaper. Wallpaper is a pattern that has translational symmetry in two independent directions. Let me talk to you about classifying patterns, wallpaper patterns by symmetries. Once you're looking at a pattern and you believe that it's a wallpaper pattern, then you wanna think about what symmetries does that pattern have? And we could try to list all of the symmetries but that is going to be a futile task because like for instance, if I find a mirror axis, do you agree that I've marked a mirror axis? If you flip about there, the pattern remains the same but you know there's a mirror axis just down to the right of that and then another one and then another one and then another one and what? You're gonna list all those? No. And then don't get me started on the ones that are perpendicular to those. Four centers. How many four centers are there in this pattern? Since we agreed that this is a token of an infinite object, there are obviously infinitely many four centers here. I cannot list them all. What else is in this pattern? I see glide reflections. Flip and slide, flip and slide and of course there are many of those. Now the way that mathematicians keep track of this infinite collection of symmetries is to use a technical concept called a group. Now if you know groups then you're all set. If you've never seen a group before my belief is that this is the best first place to learn about this mathematical concept. It's sort of a generalization of adding numbers, right? You add two numbers, you get another number except here the two things that I'm putting together are transformations of the plane. So I'm gonna transform once and the pattern didn't change and I'm gonna transform again and the pattern didn't change. Well when I transform the first time and then the second time in that composition of those two symmetries did the pattern change? No, duh. Therefore the symmetries of this pattern are closed under composition. A really important idea about the concept of group. A group has to have what's called an identity element like adding zero to any number returns the same number. Why is the identity transformation a symmetry of every pattern? If you don't do anything, does the pattern stay the same? Oh yeah, okay. So this does, oh and inverses. If you do something and the pattern stays the same and you undo that same thing and the pattern stays the same, oh yeah. So the inverse of a symmetry is also a symmetry and these are the rules that you need to make the symmetries of a pattern form what is a structure that is called a group. Now groups have a lot of structure and properties we can exploit but the idea is that this in a way is going to limit our freedom because once you decide I wanna have this symmetry and I wanna have this symmetry then you are forced to have all the symmetries that come as compositions of those symmetries. You put in two ingredients, you gotta get all the ingredients that tumble down by composing those two that you chose. Freedom and constraint, doesn't that fit with that theme? So I wanna help you with some ideas about identifying which group of symmetries correspond to this pattern. So I'm gonna hope that you will leave tonight knowing more about how to draw what's called a fundamental cell. So and where I'm gonna show you a few examples and then ask you to try it, okay? So here's what you do. You have to start by identifying the highest center of rotational symmetry. So in this case that's four. There are also two centers here that I didn't mention but the highest degree of rotational symmetry here is four-fold symmetry so I will identify that and put a yellow square there. Doesn't that seem natural a square to mean four-fold rotational symmetry? Yeah, so then I'm gonna try to get, I wanna make a parallelogram whose corners are translates of that yellow square. Now you might think that you could go down to the right and get a translation but no, that's not a translation. Here we have a spiral going that way and here we have a spiral going the other way. So if you try that little diagonal rotation that is not a symmetry of this pattern. Instead you have to locate your nearby four center there and it's pretty natural there and there. When I identify these points I have identified a fundamental region so that if I really did wanna do the stamp, stamp, stamp thing I could take that square and stamp out the entirety of the pattern. So that's your first job is to identify the boundary of a fundamental cell using as corners the places where there is the most symmetry possible. So that's a rule. Most symmetry at the corners. Then you start decorating your cell with other features. So what did we see here? We saw mirror symmetry. So I'm gonna put these two little parallel lines. Two parallel lines indicate mirrors. That's a convention. It's a very widely adopted convention and I don't draw the whole line going all the way up and down. I just draw the piece of those parallel lines that meet the boundary that are inside the fundamental cell. So I draw that and then dotted lines are gonna be glide reflections so it'd be dotted lines and I was lazy I only did two of the one of the four pieces of mirror axis that are in the fundamental cell. But anyway, make a long story short, you do that. So I'm gonna be asking you to do this and it's not that hard to put in the mirrors, put in dotted line for the glide reflections. Let me show you the cell diagram without the wallpaper. But once you have the diagram you look it up in a table of cells that I will give you and say look it's a P4G pattern. I have classified it. Here's what the fundamental cell diagram looks like. So I think you'll recognize the two centers are diamonds. Doesn't that make sense? Diamonds for two centers, four centers. There's that different four center in the middle. That four center in the middle is not related by translation so it can't be the corner of a cell. And I've got these glide reflections. Now the power of the group concept is you can take just a few of these symmetries and combine them and generate a whole bunch of other symmetries. And this group has a very special property but let me tell you about its name. It's called P4G. The P means primitive cell which is to distinguish it from a centered cell which I don't wanna tell you about right now. The four is for four fold symmetry. The G is for the glide reflections. So this is called P4G. And this is the vocabulary of the International Union of Crystallographers. And they really are studying this stuff because this is the reality of how matter is made. Matter has some of these different kinds of symmetries. So crystallographers needed language and this is the name they chose for this particular set of symmetries. And the thing that is a little bit mind-boggling and is your exercise if you wanna go home and do some math is that that entire group, all those infinity of four fold rotations of translations, everything can be generated if you just have one mirror axis and a four center that is not on the mirror axis. If you've got those two things, you can put them together to make everything in that group. Fantastic. An infinite group generated just by these two little things. Yeah, then if you wanna get started on your exercise if you rotate then mirror, then inverse rotate then mirror, you don't get back to where you started from, you get somewhere interesting. So go home and try that. So two patterns we're gonna say they have the same type if they have the same symmetry group. And by that I really mean their symmetry groups are isomorphic. Okay, it means the same except maybe turning or expanding or something like that. So your first quiz question here is are these the same or different? And I would like to be able to enforce a rule like I can in my class that everyone has to vote. But take a minute, look at those, are these the same? Are they different? How many people vote for the same? How many people vote for different? I tried to fool you because I turned the fundamental cell on its side so the right hand one has a pretty different visual appearance. But look, it's got a mirror axis and it's got fourfold rotational symmetry that is not on that mirror axis and those two symmetries together we said are known to generate all of P4G. Wow, these are the same. Okay, so your first work out, yeah? You're pointing out something very interesting which is that two patterns that have the same type meaning the same mathematical classification can indeed look very different so that this has very different kinds of shapes than this one. But according to the mathematical classification by symmetry group, these patterns are said to have the same type. But yes, I very much agree with you, they are visually different. Now, I want you to look at these two patterns and not just think do these look pretty different but I want you to think about the math question are they the same pattern type or do they have different pattern types? So you've had a minute to look. Who votes for same pattern type? Who votes for different pattern type? The argument in favor of same pattern type is that yes, each pattern has three centers and mirrors. Good, you have recognized that there are three centers and mirror symmetries enjoyed by both patterns. But on the left, if you pick any one of these three centers and think about the symmetries that preserve that point, that's called the point group. It's a subgroup of the group of symmetries. On the left, every single point group that contains a three center also has mirrors. Whereas on the left, and that's called P3M1, whereas on the right, look at those blue points there. Those are rotors with no mirror symmetry through them. So on the right, some subgroup that fixes a point has no mirror axis in it, and this one's called P31M. So this is where I apologize to you all on behalf of the International Union of Crystallographers because these are terrible names. But I've learned that M1 means more mirrors and 1M means fewer mirrors, so okay, fine. To tell you about the photographic material here, this is Point Lobos on the left, and shows that it can often be the textures of a photograph that are more important than the colors in making a beautiful pattern. And then the spring flowers from San Jose turned into bright daffodil colors there. There is a great wallpaper surprise, and if you don't know it already, I'm glad to be the person who tells it to you. How many different pattern types are there? There are exactly 17 of them. Wow! Who makes this stuff up? I know. This limitation is part of the nature of our reality. It's very much connected to the nature of matter, the possibilities of matter that can exist in our universe. So I think it's a pretty interesting fact that there are exactly 17 of these. This is my diagram of these. I use yellow diamonds, which may be difficult to see in the handout, but so if you were looking at this piece of paper and you had drawn that P4G cell diagram, you would look at this and say, oh, I see, it's P4G. I've got a P4G pattern here, and I can now answer that question that I asked at the very beginning, which was, can I be so creative as to have six-fold symmetry but no three-fold or two-fold symmetry? Well, in the bottom row here, these are the only wallpaper groups with six-fold symmetry in them, and they have translation, and you might not be able to see, but every one of these has a triangle in it, as well as a diamond, showing that if you're gonna have six-fold symmetry as well as translational symmetry in even one direction and therefore two, thou shalt have three-fold and two-fold rotational symmetry. No one can create a pattern that is truly wallpaper that breaks one of those rules. Another example of freedom and constraint, and another time when I say, who makes this stuff up, right? So I find this a very beautiful and moving fact, the 17-ness of these, and some years ago, I celebrated that in the Chuck Slam at Carleton College. Carleton had just repurposed it, purchased their local middle school and repurposed it to an art building. They pleased all the mathematicians by preserving the beautiful old chalkboards there in Northfield, Minnesota, and so they had a Chuck Slam where I celebrated, it took me about two hours to draw the 17 diagrams for the cells of the wallpaper groups, and at the time I had an exhibition of my mathematical art running at Carleton, so I could point and say, see this one there, you go down the hall, you'll find one of those. Once you start seeing these, you will see them everywhere, I promise you, and this is from a very famous movie, so you're watching the movie and you say, oh wait, what pattern is that? And very quickly, I'll take you through the classification. See those hexagons there? Does that mean there's six-fold rotational symmetry? No, the hexagons are kind of fooling you because if you turn by 60 degrees, those vertical stripes turn that way. So no six-fold symmetry, there's two-fold symmetry, two-fold rotational symmetry about many points, for instance, here, and then I've gotta get a translate of that, this is not a translate of that over there, oh good, there's a translate of it. So I go through and I draw the fundamental cell with its corners at the centers of two-fold rotational symmetry, I get that. The parallel lines show the mirror axes, yeah, that's symmetric that way, and then the horizontal dotted lines show the glide reflections. So I look that up in the table, that's called P4G. Now while I've been talking about it, how many people have recognized the movie that this is from? From The Shining. It's from The Shining. That's shocking. It's iconic, isn't it? Yeah, and there are several other patterns in that movie that are like scary, and I don't know if Kubrick really planned it or if he just, yeah, okay, saw something that was kind of amazing. So like I say, one of my learning goals for you is I want you to become the person who pauses the movie to look at the wallpaper. Please become that person and write me and tell me that you're doing it and tell me movies that you've found things in. And as you just go about through your daily life, so last night I was at the Ethel Barrymore Theater to see the band's visit, and I looked at the carpet and I thought, look at that. And so I had to annoy the usher who was trying to clear the theater and get a photograph of that. And I'll quickly do the classification for you. Two centers where the green leaves meet two center, two center where those curly things are, yeah? It's a PMG, you recognize it, you recognize that cell diagram. So do you think that we're the only people in the world who really know that the pattern type of the carpet at the Ethel Barrymore is the same as that weird pattern in The Shining? You know, that's unique knowledge. But so as you become a person who sees these everywhere, you will start recognizing these pattern types. There's a beautiful set of woodcuts over there that you can see on your way out, find the pattern types in those. So yeah. I could say that this is a different representation of PMG because in the first one, the mirrors were this way and the second one, the mirrors were this way. And that's what I mean by isomorphic groups that yeah, you can turn this one to be that one. There's a technical definition of group representation that fits this. Let's do one more example, that's gonna be your turn. So I grew these sweet peas, I've got more on the way, use my process and turn them into this pattern. There's sort of a weird feature of this pattern that okay, I've identified the center of two-fold rotational symmetry. These little white figures are the same up to down. If you turn them upside down, the pattern is the same. But as I go to fill out my fundamental cell, I might be tempted to go all the way over there, but that is not a shortest translation. To get a shortest one, I have to go up there and then down there. And this makes the characteristic pattern of what I call the centered cell. So this is a diamond. So you might see a pattern in your classification exercise where yeah, there's a diamond there, it just jumps out at you. Quilters call this a half-drop repeat pattern because it's like I've taken the row of white figures, translated it and then dropped it by half. Half-drop repeat, I don't know if anyone has heard that phrase before. Well, I identify the mirror here, there's another mirror perpendicular to that one. And when I fill in the cell, I get this diamond with the mirror axes through and did you see the glide reflections before? There are glide reflections there, flip and slide, flip and slide, those always will happen in this centered cell pattern. So I look at that and say, yeah, that's a CMM I've got there and I enjoy it more for understanding its mathematical classification. Yes. By its group theoretic mathematical classification, the C is for the centered cell. PMG is different and you can't have a CMG, it turns out. PMG is one thing, a CMM is totally different. They're connected. So now it's your turn. I'm going to enlist the help of some docents. They have two things. First, the classification diagram and some advice about classifying things and then your first challenge one which has two patterns on it. If you can identify those two, then somebody can bring you a second and a third. So we have six patterns. We're going to take about, I don't know, maybe 12 minutes for you to try your hand at classifying patterns. This gives a quick introduction into how to do the fundamental cell on the back. On the back, there is the cell diagrams. Then your challenge here. The challenge then is to, even if all you have is this one, you can start trying to draw a fundamental cell on it. Can you get a fundamental cell drawn on here? What's the rule? You find the centers of the highest rotational symmetry and then try to make a parallelogram that connects those. If anyone wants help with their classification and if anyone has finished two classifications, you can ask for a new page. That's correct. That's correct. Sorry about that. Yep. No, because there's no six. I see what you're seeing by the sixes, but there's no six there. So it's more like this because through every three center, there are mirror axes. Okay, I think that we'd like to move on. If you didn't get challenge three, I don't think anybody got challenge three, but it's up here waiting for you to enjoy. There are also a limited number of cheat sheets with the answer key that we prepared for our docents to help. So if you'd like to steal one of those or photograph one of those, that would be great. So did you have fun looking at the patterns and trying to recognize their features? There's something about that that is an inherently pleasurable human activity. So build that in to your visual vocabulary. I would like to tell you a little bit about how these images were made and to try to explain to you the concept of what a wallpaper wave is. Well, the symmetry work software allows you to put in any photograph and manipulate these wallpaper waves and you could turn this photograph into that pattern. It's very flexible software, it's open source, it might be a little kluji, might crash here and there, but it's free. The basic ingredient in symmetry works and really in all of my work is the concept of a plain wave. Now I want you to think of waves not as a wave crashing on the beach, but a wave in the deep ocean that is just smoothly undulating, always the same, up and down, up and down. I've depicted that with color. As if I don't know, maybe it's red, high and yellow, low or no, cyan low, that's what it would be. So as we go through red to green to blue that just those colors just represent up and down, up and down, always the same. Well, this does have translational symmetry in two different directions and I'm gonna think of this as being a plain wave that shows the hues in a color wheel. So in my early work I was kind of obsessed with color wheels and wanted to think of the correct way to construct a color wheel. Here is just one example. I tried things with infinitely fine gradations but I found it was more productive to have some breaks for the eye, to see specific sectors. So even though I said I don't like things broken into pieces, I have spent a lot of time breaking a color wheel into pieces to give visual variety. I'm using the Euler formula, every mathematician's famous formula, favorite formula, yeah. This is E to the I, Y. So as Y goes up, this function goes around. So this is just a way of telling the plane there to be colored with the perfect hues. That is the formula for a plain wave. In one of my adventures with color wheels I invented, oh yeah, so I'm gonna say complex number but you can think color if you'd rather. So I've just made an identification. A color is the same thing as a complex number. So I've made a lot of color wheels and this one I commissioned glass artist Hans Schepker to construct this image for my house waiting for me back in San Jose. It's really beautiful and it has 22 sectors of different colors. Later I'll be talking about doubling the frequency. That's a wave that goes twice as fast. Doesn't that look like the one on the right is twice as fast? Yeah, so this is what I mean by a higher frequency wave, one that goes maybe twice as fast. As I think about doing algebra or multiplications with waves, the one on the left is multiplying by one half because I was having those colors sort of fade to white at the center. So if I go half of that wave it's gonna be colored this way. If I multiply it by two it's gonna be out in the darker region of that color wheel. This is a remarkable diagram and one that really delighted me the first time I saw it. What I'm doing here is taking that basic plain wave going this way. I'm taking another wave that is offset 120 degrees and a third wave that is offset 120 degrees from that. So I've got three waves going like this and you would think that they would just create chaos. Those three waves going in three different directions but they form exactly this diagram as long as you add the colors like complex numbers. So it's not such a stretch to think that you could add colors after all if you have a red light, a blue light and a green light and shine them all together as in this diagram you'll get white light at the place where they all overlap. And that's what's responsible for the white dots in this diagram, in this on the left that's places where the waves come together so that there's just as much red as green as blue. And that combination of waves creates symmetry. What was your question? Why would it be blue and yellow? Well you're maybe thinking of a different set of primary colors in lights, the red, the green and the blue are the primaries as you'll see if you look really closely at your computer screen. Hence the RGB color system for controlling red, green and blue pixels on the screen. So this is a real clue as to why these things work and it's, I don't know, is it because this group exists to begin with that you can do this with the waves? For me it still seems somewhat miraculous that you can superimpose these three waves and get a pattern that has both the translational symmetry of the waves and the rotational symmetry that I built in by sending these waves in rotationally symmetric directions. If you like formulas, this is the formula for that wave that I had. The first is the simple Euler formula for a wave going up this way. Then the second and third terms use the rotation formulas. So if you know your formulas for rotation in the plane that I've rotated once, twice by 120 degrees and I lock these three waves together to make a new wave that is invariant under 120 degree rotation. The formula at each point, so in every one of these diagrams you see the formula at each point gives a complex number and then that complex number is the same as a color. So that's how this diagram is made. For each pixel in this, I use C++ to say for that pixel compute the value of that expression, turn that into a color, put that color at that pixel. In symmetry works this is called the one zero wave. It's the base wave, the one with the lowest frequency. You are free to then superimpose that wave packet with other wave packets and make more and more decorated patterns. Like for instance here, I made a wallpaper function with superimposing many waves, I don't remember how many, but coloring them with this color wheel to make a pattern of type P3 that has three-fold rotational symmetry but no mirror symmetry. So the sum of waves formula again returns a complex number at each point then that complex number is interpreted as a color. So this was the state of my art in the late 90s was making patterns using color wheels. And I drove myself a little crazy inventing more and more color wheels, more and more algorithms to associate complex numbers with colors. And then I took a big step in the direction of art which was to say color wheel, the world is my color wheel. I can just color the complex plane like that. That's just as good a way to color the complex plane. And then when I use that as a color wheel with a function with the right symmetry, then I get a pattern. That's where the art started to get really fun for me. And the potential of using the SymmetryWorks software it's huge. I would love to think that out in this audience there is somebody who's gonna make better patterns than I have. I think I've made some pretty darn good patterns. I like the patterns that I've made. I've made them to please myself, but everyone has different artistic sensibilities. So why should you not be the person who makes the best artwork with this kind of idea? I would love to see the patterns that you will make. Here's how the typical screenshot of SymmetryWorks, you can get a pretty nice preview here. There are things, lots of controls. You can pick the wallpaper group you wanna use. You can make more of the pattern or less of the pattern. And then there are these really cool scaling devices where in my old version, I had to type in the formula for a complex number. But here you can pull up a screen and pull a control around a plane because that's what you're doing is picking a point in the plane to be the coefficient of one of the waves. It's very fun. So what makes it art? Well, I say it's my artistic choice. This is a quote from a guest editorial on the Journal of Mathematics and the Arts. The Journal of Mathematics and the Arts, isn't it wonderful that we live in a time when there is such a journal? So they're talking about what makes a computed image art? It's the person who has sifted through hundreds, if not thousands of images looking for art. So that's what I think my art is, is the choice of which of these infinite dimensional, which point in this largely infinite dimensional space will make a picture that I think is worth sharing with others. Scientific American asked me to do a guest blog. They wanted a wallpaper that had their logo in it. So I figured out it was hard to make that happen. But so this surfer on our California coast turning into this thing with the glide reflections. You see those glides going up like that? Here's some favorites of mine. This comes from a photograph with almost no color variation in it. And it makes quite a minimalist pattern. But I've been in several situations where I've asked people to select from a collection of a dozen or so of my wallpapers. Say you want some wallpaper? Yeah, I've got some prints. I could send you something for your exhibit. And this keeps being chosen again and again. People seem to really like it. Maybe it's because they recognize P4G. And that's a favorite pattern type of mine. You can be funny with this. If you use figures, they get pretty weird, pretty fast. But I took this picture of myself. I was cutting beats and caught myself red-handed. And then made this pattern. Now a reason to look at this pattern is that this has the least symmetry possible for wallpapers. There is only translational symmetry. No rotations, no mirrors, nothing. It's just repetition again and again. And I tend to like the patterns with more symmetry. But this is beautiful as well. Now I don't want to take too much time, but there's a whole nother thing you can do with color-reversing symmetry. Where I took this rather homely photograph of an early spring scene in Minnesota. And I made it into a collage by taking its rotated negative. And then I combined that with waves in such a way as to create color-reversing symmetry. Take one of those diagonal axes. If you reflect about one of those, all of the colors turn into their exact photographic negatives. Color-reversing symmetry. You really will see this. I still remember the first time I was in a restaurant and I looked at the wallpaper on the wall and thought color-reversing symmetry. It really is out there. Really is used. Here's another example made from my same stained glass window, except I had to also use a collage with the stained glass window and its negative. Here's an arrow pointing at a color-reversing, well, it's a three-center. That's a three-center when you think of actual symmetry. But if you rotate by only 60 degrees through that point, every color changes to its exact opposite. Color-reversing symmetry creates a striking effect. This is a favorite pattern of mine. So everyone wants to know, well, how many ways can you do that? Because mathematicians classify things, count the number of ways to do things. There are 17 wallpapers and there are 46 color-reversing types. This fact was first published in the Journal of Textiles, Manchester, in 1936. Why did they want to know? They were pioneering jocard looms where you could easily program a weaving machine to make fabric and they want to know how many two-color fabrics they could create. So it's a long adventure with color-reversing symmetry, but I'd like to move on and tell you about the symmetry variations, the things that have happened to me after wallpaper. Morphing wallpaper, which is what you see there, spiral wallpaper, non-Euclidean wallpaper, imaginary landscapes, and of course, if wallpaper is made from waves, waves know how to move into the future, so you have to have vibrating wallpaper. Now our time is short, so I'll be brief. In this symmetry variation, this is connected to the pattern on my book cover. It was printed four feet wide and nine feet tall on fabric for an exhibition at Carleton College. It's quite beautiful. I took this picture of a peach, reversed it, and rotated by 180 degrees. Then at the left-hand side of this image, this is a close-up of the left-hand side. There are these blue, flowery things, and then peaches by threes. So do you see three centers that are peachy and six centers that are blue? I started with a pattern so that at the left-hand edge, that was the pattern, and then I gradually turned the photograph perfectly upside down until I got to the far edge, where now, because the photograph is upside down, I am seeing peachy six flowers and blue peaches by threes. So that's called a morphing freeze. It's really fun to do that with fabric. So here's a beautiful fabric. I've got a site at Spoonflower. My designer name is Frank Ferris. There's some fabrics there. I wish I had my yard of this beautiful fabric to wave at you. But I'd like to spend a little bit more time telling you about spirals. At Bridges last summer, our wonderful conference of mathematics and the arts, I met John Edmark, who gave a wonderful talk about his spirals. If you don't know about his Blooms series, then I encourage you to Google him. I'm told the museum is going to acquire three of John Edmark's Blooms. Well, I took his work and realized that I had all the tools to make my own blooming things. And if you know a little complex variables, you know the complex exponential map takes the plane and winds it around in a spiral. So that's something that is known to people. You can take the plane, wind it around in the spiral. If you wind it just right, then your wallpaper pattern will match perfectly. So it takes, you know, the mathematical story is interesting, but so I wound this wallpaper pattern. Does everyone see the P31M there? No, you need more time. And I want to just mention, I love the Sierra gooseberry. My family has been making jelly out of these gooseberries for generations. They're weird to pick because they have thorns all over them, unlike your tame Eastern gooseberries that you get here. So Edmark's Blooms series uses a property of Fibonacci spirals. So look at that sort of white flowery thing at the top of that yellow arrow. If I rotate by exactly the right angle, which is about 135 degrees, called the golden angle, I get another copy of that same figure only enlarged a little bit. So if I were to strobe this, your eye would see that that smaller thing had turned into that larger thing. And John Edmark's Blooms do this with strobing. I do this virtually by just computing a movie where the next frame is going to be the rotation of the previous frame by that special angle, resulting in this kind of wonderful blooming effect. But then I took it a step further, being liberated from the idea of a physical thing that gets strobed. I realized that if you think of a complex plane as being the same as the Riemann sphere, anybody know that? You take the sphere, the plane, you pop it up onto the sphere, and that means that the pattern that's coming out of here is going into there. A pattern is emerging from here and going into there. Well, and then there's a common way to take both of those ins and outs and just stick them down where we can see them, bring the North Pole to a finite point. The strobing effect still works. And that's my idea of a good time. But then I said, wait, there's more. You can actually compose this with any complex mapping. I'm gonna show you one that has two sources and two sinks made from the same technique as Edmark's Blooms, but more heavily involved with complex variables and less connected to anything you could realize physically. It has sort of a summer of love vibe to it, right? I'll just briefly mention non-Euclidean wallpaper. It's a whole other story. Have me back to give an hour talk about that. Here just the gestalt is that these pale blue triangles are all the same size going down into the edge of the universe. And so it's a whole story about creating wallpaper there. But I always have to say these are not fractals. There might be a fractal dust around the edge there, but I come from the land of smooth. These are smooth things. Yeah, polyhedral symmetry, you play the same game. You find functions on the sphere that are invariant under the symmetry groups. This one is invariant under the icosahedron made from my same window. Got a lot of mileage out of that photograph. But this has a singularity, my original technique and the technique that's in the book requires these singularities. And I had to sort of pinch the photograph together there. But Jeff Weeks and I have a joint project. He taught me how to get rid of the singularities, but I had to use an image that colors the entire sphere. So I took a picture with a spherical camera. If we were able to view this correctly, you'd be able to look all around the dome of the Lakewood Chapel in Minneapolis. It's a beautiful building. That mosaic goes all the way around a circle above you and with the right functions that turns into this polyhedral-symmetric object. So then I made this piece called the Alchemist's Shelf that has six globes with different kinds of polyhedral symmetry. Four of those are different photographs from Minnesota and the top left and the bottom right are of the stairwell in my house where the window is. What about this opening slide? Well, I'll talk just for a couple minutes about my imaginary landscape series. Two years ago, I updated my Photoshop and I saw a button that said 3D. So like a good primate, I pushed the button and the rest is history. It enables you to do fantastic things. These things that we are all able to do in Photoshop were only available to people who worked at Pixar and Dreamscape five years ago, but now we can all do it. So again, this is colored with my picture of the window. It's by a technique called ray tracing where it does the computations to pretend that there is a camera there and traces all the rays. The first thing you do is you create virtual three-dimensional objects, then you paint them with patterns and I had to do the computations, figure out how to put the pattern on a sphere. You tell them what material they're made of, so I told that vase, by the way, be made out of crystal and ball be a little bit transparent and then you can shine lights out of them. You can shine lights on them, shine lights through them and you can see the back splash of the light on the reflective surfaces that I put there. I'm dazzled that these kinds of things are available to us. This is a fanciful view of five globes landing on a lake up in Sierra Nevada. They arrive starry night as if the stars in the sky are in a giant spiral pattern that is reflected in the lake. But back to wallpaper, two Sunday nights ago, the San Jose Chamber Orchestra commissioned a piece. They asked a composer to write music to accompany my movies of vibrating wallpaper. It came in five acts, each with a different photograph of a scene in California. The piece is called In A State of Patterns and is an homage to California from the beautiful poppy to life in San Jose to the Sierra Tree death and then to the stars. So this is the score, minimalist composition, it's one minute long, and then we will rise. That synthesized music, the real orchestra playing was far more beautiful and had five acts. So I've been very moved that these many projects have come into my life. But you can see that I've found just endless possibilities here. And I don't want to think that I'm the only person working in this art form. So what my hope is that people will take up this strategy for producing patterns and run with it and make things that are far more beautiful than I could create. So my question for you is, what will you create? Thank you. All right, we have time for I think just a couple of questions. So if you have a question, you can raise your hand and I'll bring the mic over. So is there some fundamental reason why there's 17 patterns? It's hard to explain, but it's just true. It's just that's how many there are. And the way you prove it is by saying, well, what could you do if you had this and run out of possibilities? There is a proof in my book, no. Many of the patterns that you creates with the symmetry seem to be reminiscent of the like magic eye stereogram type things. Is there a relationship between those? The magic eye relies on patterns that are not periodic. They change so subtly. So when you look at them without viewing them the correct way, they appear to be sort of periodic. But it's the fact that the two pieces don't quite match that makes the magic eye things pop out. When I started doing this, I thought, oh, this is gonna enable me to make great magic eye stuff, but I think I was just wrong about that. Let's give our another hand to Frank Fez.