 I'm Marietta, I'm from the undergraduate program, and our goal was just to create the 17 different types of wallpaper groups, and we really wanted to work on making them look artistic and interesting. Yeah, I am Betty and I am a high school teacher in Los Angeles, and one of the big topics that I do teach to my students is transformations. And when I was at Mr. or Professor, if I was to talk last week, when you talked about different symmetries and how you can make different wallpapers, and I thought, well, my students always wonder about when are we ever going to use this symmetry and all this transformations, so I thought it would be really cool to make some beautiful wallpapers and share it with my students. Okay, so like I said, we wanted to make the 17 different types of wallpapers. The image on the left side is the different cells for the different types of wallpapers, so that's a super helpful tool when you're trying to identify which wallpaper is which, and then on the other side we have a flow chart that we found which targets unique features or unique combinations of features of each wallpaper to help you narrow down what wallpaper you're looking at, because it's pretty easy to go into symmetry works and make a picture that looks cool, but it's a lot harder to identify what it is, and you can actually make things that aren't any of the 17 patterns, so the whole goal is how do I identify that, and yeah. Okay, so again, these are the 17 cells just to explain a little bit. The solid black lines just border the cell, they don't tell us anything about the symmetry of the cell, they just make the border, so then the yellow shapes represent rotational symmetry, so yellow diamonds represent two-fold rotational symmetry, triangles three-fold, squares four-fold, and then hexagon six-fold, and then the blue parallel lines, those represent mirror lines, and so the axis at which the image mirrors actually happens right between those two blue lines, and then the green dotted lines show glide reflections, so all in all you have rotational symmetry, mirrors, and glide reflections. And so we're just going to walk through how to start drawing out your fundamental cell. So, first rule is that if your pattern has any rotational symmetry, the corners of your cell must appear at the centers where there's the highest rotational symmetry. So for this wallpaper, the highest rotational symmetry we have is three-fold symmetry, so I'm going to choose the corners of the cell to appear here. We also have three-fold rotational symmetry in the center there, and the border of our cell, that's going to be that line of our fundamental cell. Next, I think it's easiest to look for reflections because those are pretty easy to pick out, so we have our first one right there, there's one that cuts diagonal across the cell, and two more, so those are all of our mirrors. Next, we'll look for glide reflections, and so a glide reflection is where you take the image, you flip it across an axis, and then you slide it along that same axis, and then you end up with the same image as you started, and we have lots of glide reflections in this particular wallpaper, I think there's eight total, and so I think we've identified all types of symmetry, and then if we check, that looks exactly like a P3M1, and so that type of pattern is a P3M1, and then so this is the wallpaper, and then on the other side is the picture where the wallpaper came from, and then those set of points show the points of the picture that were used to create that wallpaper, and so depending on what type of wallpaper you're trying to create, you'll get a different shape like this, and Frank talked to us a little bit about how he has no way right now of predicting what shape that will take, so he said that's a pretty fun open-ended question that he's going to try to tackle eventually, but so when you're trying to make the particular type of wallpaper we did, you can move around that shape and get different parts of the picture, you can't really manipulate that shape of points, and so a big question is why? Alright, so the next one is this pattern, so we're also going to choose a cell unit, so here I draw a square, and we're going to use a flow chart to determine which direction to go to, so at the end we'll determine which symmetry that was, so we're first going to ask, what is the rotation order of this pattern? So the rotation order is when you choose a corner, and how much angle, the smallest angle that you can rotate so that you will come to the exact same image, so if you look at the corner of the square, so we're going to look at these four corners, and you can notice that when I rotate that about 90 degrees, I will come to the exact same image, hence we can determine 360 divided by 90, we can say the rotation order is 4, and then we're going to determine is there any reflection, and as you can see, yes, we can fold it halfway vertically and also horizontally, and also diagonally as well. So the next question is, are there mirror lines or the line of reflections, are they intersecting at 45 degrees, and yes, if you look at those two adjacent lines, you can see that they're intersecting at 45 degrees, so that was another yes, and so we can determine that that was a P4 and wallpaper. And those are another corners of the pattern that you can rotate about 180 degrees, which is rotation order of two. There are some glass symmetries that I didn't include, but there are some glass symmetries as well. So here's P4M, and does anyone want to guess how or which picture I used to create this pattern? It's pretty random, but... All right, so remember two days ago, we got a bag of free chips, so I just used that image when I was here. Two days ago, and I created a beautiful pattern. So here's another example of P4M. So when I got here, I went to a lot of hiking places, and I took a lot of pictures of wildflowers. So I took one, and that became a beautiful paper pattern. There's another one that says P3M1, and does anyone want to guess which image I used for this one? Yeah, pretty close. So I went to this cafe in LA, and they made some beautiful shaved ice with some flour on it. It's like, why don't I use that? So that's P3M1. And these are some of the pictures or different wallpapers that I have created. This one is made from some avocado. And I think this one is pretty easy to guess. Does anyone want to just take a chance? Yes, that's right. So remember we walked on 4th of July, we took a picture that became a pattern, and those two, this one and the other one, were from wildflowers during my hike. And this one was also from wildflower, and this one was from Lake Tahoe. Okay, so real quick, we're actually going to do a game. So if you guess the image, you can go home with one of our wallpapers. So this one's a pretty tough one, but does anyone have a guess? Give you a hint, it's in nature. Yeah, it's a tree. Okay, next one, this one's pretty easy. So first one who gets it, yeah. So if you see my water bottle. And it's a pretty low quality picture, so you can make really cool patterns out of not that great pictures. And then the last one, a little bit trickier. No, it's a fruit. Great fruit, yeah. So yeah, we had a lot of fun with this project. We'd encourage everyone to download Symmetry Works, play around with it, make some cool patterns. Thanks.