 Hello, everyone. I'm Yamna. So in this talk, we're going to learn about Islamic art, math, and how to use tiling theory to make some pretty patterns. So before we get into the math, we need some historical and cultural context. So the rise of Islamic art began around 1300 years ago, as Islam was spreading throughout the Middle East and Europe. In general, Islamic art encompasses much more than explicitly religious art, as art that comes out of Islamic societies isn't necessarily inspired by Islam or created by Muslims. So early on in Islam, many Muslim scholars were worried about idol worship, and so they tried to strongly discourage the depiction of any living beings from their art. So that meant that art could no longer encompass or incorporate humans or animals, and in some extreme cases, even plants. So you think that banning living beings from your visual art would result in very boring art, but the exact opposite happened. Islamic geometric art and Arabic calligraphy rapidly became very sophisticated forms of art. And in fact, one interesting loophole that calligraphers came up with is that they'd shape their calligraphy in the shape of animals, so that then they could argue that they weren't actually representing animals, but rather that they were just some letters. So because the construction techniques were these very complex geometric pieces were considered trade secrets, the knowledge of how to make them was for the most part never made public. Thus we don't know how a lot of tidings and patterns from that era were constructed. We know that in general all they had to work with was a compass and straightedge, and that they had a solid understanding of planar geometry as was written down in Euclid's book, The Elements. But trying to reverse engineer their construction methods is actually still an active area of research. Anyway, oh wait, so this is from the ceiling of a mosque in Iran on the left and on the right, I have no idea where this calligraphy originated from, but it shows up in every talk on Islamic art, so I had to include it. And before I move on, I just wanted to point out that representing living beings in Islamic art is no longer mostly frowned upon in most cases, and that Islamic art is still alive and well, and it's awesome, and there's like tons of amazing Islamic art out there. So the kind of Islamic geometric art I'll be focusing on is what's referred to as star patterns, and these examples in green are star patterns. There's no precise definition for what a star pattern is, but they're said to be based on star polygons, which is just a fancy way of saying star-like shapes. And although there's no precise definition of what star patterns are, there are many, many ways to classify star motifs in the patterns they make up, and actually lots of rigorous math behind them that I don't have time to get into in this talk, that's like a whole other talk. Actually, if I'm honest, it's more like a 16-week lecture series. So what is tiling theory? At a very high level, tiling theory is the study of tiling the plane with certain kinds of shapes. I know that's really vague and we'll get more precise soon, but here are some examples of tiling you may be familiar with. So there's the very famous Penrose tiling. There's the recently discovered 15th pentagonal tiling that was in the news a few years ago, and then there's the coolest monohedral tiling that exists. This one's not actually famous, but I really like it, and I think it should be famous. So how do we precisely define a tiling? Well, here we have a definition, and on the left there's the definition for mathematicians, and on the right we have the definition for everyone else. So I'm going to read the one on the right. So there's only really four rules that tilings need to adhere to. So a tiling is a set of tiles with the following requirements. You can't have any holes in your tiles, and you have to tile the entire plane, and you can't leave any points out. Number three is you can't have overlapping tiles, and number four there are fixed upper and lower bounds on the size of the tiles, and I like to think of number four as the no fractals rule. So part of classifying tiling, we say that a tiling is vertex transitive or isogonal. If all of the tiling vertices are the same, and what that means is every vertex in the tiling is surrounded by the same kinds of face and angles in the same or reverse order. So we see this tiling on the left is vertex transitive, because every vertex is surrounded by a face of degree 4, 6, or 12, but that's not the case with the tiling on the right. And if a tiling is vertex transitive, we can give it a naming convention as follows. Where the naming convention is just the numbers, or just the degrees of the faces of every tiling vertex, so in this example it would be 312 squared or 31212. So how do we actually go from tilings to star patterns? Well there's a method called Hankin's Polygons in Contact algorithm, and as far as we're aware of it was first written down by this guy named Ernest Hanbury-Hankin in the 1920s. I don't want to say it was invented or discovered by him because we don't actually know how a lot of tilings were constructed historically. And one interesting fact about Hankin is that he was actually a biologist and he studied these patterns in his free time. And now where he did as a hobby has had a significant impact on research mathematics in this really specific field, which is something I like to keep in mind whenever someone comes up to me and is like, why do you spend all your free time reading math papers? So before we get to the algorithm, we need to familiarize ourselves with some notation. So pretend that this is an arbitrary tile in some tiling you're working with. So along any edge we have a midpoint M and from M we will have two rays emanating, the left ray and the right ray, and both of these rays will be separated from the edge of the tiling by a contact angle referred to as theta. Okay, so step one we have to pick a tiling. We're going to be using the honeycomb tiling or using the naming convention from before also known as 666. So we're going to be focusing on one tile for simplicity. So first we identify the midpoints of the edges of the tile and then we draw two rays eliminating from the midpoints. I only did one to keep the diagram simple and in this angle theta is going to be about 60 degrees. And so you would have been doing that from all the midpoints at once and then the lines are continued until they meet other lines of similar origin and then you stop when you meet the lines. And that's how you end up with this six pointed star. And then you repeat that for the entire tiling and then you remove your original construction lines and now you have a nice star pattern. So that's basically the algorithm. Okay, now we, so I have an implementation of this and I can't see. So here's the one I just showed you and we can remove the construction lines and we can look at other tilings. So here's 4612, which I think is a bit more interesting than 666. That's my favorite. So the algorithm that I described doesn't actually work for non-regular tiles reliably. So what ends up happening in this particular example is the lines end up meeting outside of the tile, which might be okay if you're just like implementing a visualization in JavaScript, but that's not going to be okay when you're hand painting clay tiles that you're going to use to decorate your mosque. So there's a really clever solution to this. So ideally what you'd want is something like on the right. And that solution uses like graph theory and exciting matching algorithms, but I don't have time to cover it. But if you're interested, I'd recommend reading this paper, Islamic Star Patterns and Polygons in Contact by Kraub Kaplan from the University of Waterloo. And by the way, most of the context in this talk is based on Kaplan's research and I highly recommend looking at his other research because he makes tons of really pretty pictures. So one interesting thing about Hankin's algorithm is that different base cut tilings can lead to the same result. So we see that we have this tiling on the left and this tiling on the right and they're different, but if you apply the algorithm with a contact angle of 54 degrees on the left and with 36 degrees on the right, you get the same star pattern emerging. And the reason for this is because there's a special kind of transformation between the tiling on the left and the tiling at the right because at a fundamental level, they have very similar symmetry structures underneath and that transformation is called the rosette transformation. I don't have time to get into it, but if you are interested, I recommend reading the paper I mentioned before. So another thing we can do with this algorithm is we can vary the contact angle at a linear rate across the canvas. And from this, we get what's referred to as Islamic parquet deformations. So a parquet deformation is a design which like smoothly evolves through space and parquet deformations were originally introduced as exercises for architecture students by the architect William Huff. And as you can see in this example, the contact angles peak in the middle, which is like why everything's a bit more angular. And so it goes up and then comes back down, which is why it's the same at the beginning and the end. And up top, you can see the individual tiles changing. And now that the idea of varying the contact angle has been introduced, here's an example where you can interactively vary the contact angle based on distance from a point and that point can be moving. And if I put my mouse here in the bottom left corner, you can almost see the original tiling in the top right corner. And this was made by someone who only wants to be identified as by, or use of X. So there's a lot of interesting physical art projects based on these ideas, like many more than I would have time to include. So here's an embroidery project I made a few years ago. It's kind of wonky in the center, but otherwise it's okay. And on the right, there's this piece made by the mathematician, Florence Turner of a beat Islamic parquet deformation called From Kepler's Star to the Night Sky. And that's all.