A decagonal number is a figurate number that extends the concept of triangular and square numbers to the decagon (a tensided polygon). However, unlike the triangular and square numbers, the patterns involved in the construction of decagonal numbers are not rotationally symmetrical. Specifically, the nth decagonal numbers counts the number of dots in a pattern of n nested decagons, all sharing a common corner, where the ith decagon in the pattern has sides made of i dots spaced one unit apart from each other. The nth decagonal number is given by the formula
The first few decagonal numbers are:
 0, 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451, 540, 637, 742, 855, 976, 1105, 1242, 1387, 1540, 1701, 1870, 2047, 2232, 2425, 2626, 2835, 3052, 3277, 3510, 3751, 4000, 4257, 4522, 4795, 5076, 5365, 5662, 5967, 6280, 6601, 6930, 7267, 7612, 7965, 8326 (sequence A001107 in the OEIS)
The nth decagonal number can also be calculated by adding the square of n to thrice the (n—1)th pronic number or, to put it algebraically, as
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✪ Computer based design of Islamic geometric patterns
Transcription
Welcome. Good Evening to all of you Thanks for braving this weather and being here. We were supposed to have a full house but we're good with all of you. So great So this is the last of our fall Studies of Islam speaker series. We will resume again in January. We are really glad to have Professor Kaplan here and my colleague will introduce him more appropriately. Good evening everybody and thank you for coming. So I just wanted to read a little biography of Professor Kaplan. I think we are ready to hear what you have to say Craig Kaplan is an associate professor in the David R. Charitan School of Computer Science at the University of Waterloo. He has a Bachelor of Math in Computer Mathematics and Computer Science from Waterloo. and an MS and PhD in computer science from the University of Washington. He studied the application of the computer graphics and mathematics to problems in art, architecture and design. He is an expert in topics such as Islamic geometric patterns and computational applications of tiling theory He is the editor of the journal mathematics and the arts and helped to organize the annual bridges conference on the arts and mathematics So help me welcome Dr. Kaplan for his talk Thanks for coming. It's really awful weather. It is a pleasure to come out and talk to a difference audience. I mean I recognize some familiar faces but an audience that hopefully will share my interests in Islamic star patterns which I can try to convince people in math that this is an exciting topic but its not something that a lot of people there study. So my research has a lot to do with trying to understand the structure of patterns, tilings, ornamental designs and then trying to understand how to use mathematics and computer science techniques to create and design and empower artists. Make artists more expressive get them to the answers that they want more quickly and more creatively. This is something I've been looking at since graduate school I mean the topic of my PhD thesis was Islamic geometric patterns and the art of MC Escher not necessarily the intersection of those things but both topics together. I don't actually study Islam. I mean its great to be here but I won't pretend that this will be a study in Islam This is more a study near Islam...in the same vicinity as Islam My work is obviously informed by Islamic art but more as an abstract style of geometric design and not because of its religious significance in particular. Not so much even because of its cultural significance although to the extent that helps me understand the patterns that I care about. I'll say a little bit about that so first of all, what am I going to talk about I'll take you through a quick tour slightly delayed and a quick tour of the work I've done on trying to understand and create Islamic star patterns I'm not going to go over all of the algorithrims I used to detail as I might as if I were giving a talk to an audience of computer scientists. I figured that that is not as relevant here. What I liked to do instead is to say a little bit about why computer science should matter at all in this context why we should care about using math and CS to solve problems in Islamic art And then after that, I'll take you through a tour of some of the things I've done with Islamic art. I can't do complete survey of the contemporary world of art based on Islamic ideas that vast. I'm not really an expert on it so Actually you are better off talking to Sahail Who both knows a lot about it and practices it herself. Well let me begin by explaining what I mean by Islamic geometric patterns. There's lots of really wonderful easily accessible examples I can point to this is the very famous tiling from the Alhambra from Spain Its actually made out of small baked tiles even the little vent sticks. Its an incredible achievement This is an abstract geometric design that you would find in the Alhambra After having studied this for many years, I finally got to visit to Alhambra in person this year So I am freshly fired up with excitement about this topic Here is one more design. This one's not executed in tile but its carved into plaster and stone Just as evidence that I was physically present, there's me taking a picture of that design Yeah it could be photoshopped, might not have been me! Might have scoured the internet to find someone that looks like me who wears the same kind of shirts designs There are both examples but of course you'll find geometric patterns on other surfaces Here's a wonderful dome from Iran. Its got a pattern. Its actually quite complex to find a geometric pattern that fits the changing curvature of a dome Its a topics I can't get to today. Although I will talk a little bit about things in 3D You will find examples carved in wood and examples taken from elements that came from mosques so here's a wonderful examples from the Victorian Albert Museum in London I was in London over the last year on sabbatical which is why I was able to visit to Persian art room many times This is from a min bar, 15th century min bar from a mosque in Cairo Another place you will see lots of wonderful examples of geometrical design are the art of books So you got the cover of beautifully illuminated Qurans. This one isin the museum of Turkish Islamic Art which I think is Istanbul Here's one more beautiful illuminated Quran You can see there is the interaction of lots of different styles of art in one place you have therigid abstract geometry of what I will call in a second as an Islamic star pattern and you also have these beautiful symmetric floral designs that fill in the gaps The floral designs are a little bit free form and less abstractly geometric We could still refer to them as Islmaic geometric patterns because they do have a lot of mathematical structure The most obvius example being that most of the designs are inside symmetric tiles which are themselves laid out symmetrically you know in fact, the world of Islamic geometric design is not just about these flat abstract star designs at all But there's also these floral abstract designs, calligraphy plays a major role in Islamic art as you probably know and some forms of Islamic calligraphy have a very rigid geometric structure and of course Mucharnas which is a form of architecture corbeling, sort of to fill in the corner underneath a dome that is very geometric and so on Textile design in rugs and there's a lot of other areas where this stuff is practiced but I am going to focus on the topic that I've studied for many years which is what I sometimes call Islamic Star Patterns as a special class of Islamic geometric patterns Why do i call them that? Well I mean hopefully you can squint at this and see the stars peeking out at you The most obvious ones are these sixteen point stars in the design There's also a wide variety of these big scattered eight point stars in two orientations which is a little unusual actually So there's the stars in isolation and these are pretty easy to construct mathematically You can construct a regular polygon, all the sides are the same length and all the angles are the same Put a point of the star in the center in the edge of every regular polygon and then take the lines and join the points bring them into the center of the polygon by some distance anyway you end up with a whole variety of stars Another way to do it if you are interested is that you can take points regularly spaced around a polygon and join them up in while skipping some points so if i put eight points around a regular octogon and join them up skipping every other point, I end up with two overlapping squares or one particular eight pointed star here i am going to go from here and skip 1,2,3,4,5,6 skip five, go here, skip five, go here interesting process to draw these stars if that's how you want to do it I might also identify these distorted five pointed stars as being star elements in this patter its not really crucial to have an exact counting of what qualifies as a star i am most interested in the presence of these very regular stars because that is the really most identifying characteristic of the star pattern so there's a whole variety of examples of these all around the world, all around the muslim world from Northern Africa, Southern Europe, all the way across into the Middle East, Persia, as far as Uzbekistan and Turkmenistan and India You can find designs like these So we have a whole wealth of examples to draw from the question is how were they constructed I mean what do we now know about how these things were drawn originally? Who have lots of examples we can look at as finished designs but as a mathemitician I want to know where they came from? What was the artist thinking? Who first came up with a given pattern? The funny thing is that we don't have a lot of understanding where these patterns came from We don't actually know a lot about how they were designed There are a couple of reasons for that One reason is that it was kind of a secret People who were practicing this art form from hundred of years ago had a lock on the market It was the province of guilds and they weren't so willing to share their trade secrets with other people there's not smoking gu, no book out there that says five hundred years ago there was a grand book of how to design star patterns That is kind of nice for a mathematical point of view because it leaves us with a puzzle that allows us to study all of the patterns that are out there in the world and ask "what kind of system can I divide as to account for the structure in those paterns? Now that being said, there are a few bits of evidence that we can point to A few interesting documents that were remain from hundreds of years ago that tell us something about how these patterns were designed One famous one was the Topkapi Scroll from Turkey This is, as far as we cn tell, a design manual that belonged to a group of designers back in the day and they would use this as a pattern resource book and if you look closely at the scroll you will see drawings that seem to have hints of how the design was produced in this case, the final star pattern is pretty much the blue lines maybe with the red lines, the red solid lines super imposed but faintly in the background you will see the doted red lines that seem to subdivide the world in a meaningful way and break up the patterns into these units that look a little bit more managable and and in some sense thats where I am headed with my technique for constructing star patterns and I will get these soon So this is kind of interesting and in fact, in other parts of the scroll, there are even fainter marks that suggest the structure of the patterns are they're actually visible marks where somebody scraped some preparatory lines into the scroll with a metal point to use as a guide so you can't even see them if you photograph the scroll you have to look at the edge on or look at a high resolution 3D scan or something to get at the real information another document that survives from the late 19th century are the Mirza Akbar Scrolls which live in the victorian albert museum in London These were brought to London from Tehran in 1876 where they were purchased by someone who was visiting on behalf of the museum Its a little bit harder to talk about the authenticity and usefulness of these works of the scrolls. Its not clear that they have any real long history the person who is the curator of this material at the museum has a theory that maybe somebody went to Tehran and said, "do you have some drawings that we could take back?" and they said, "yeah come back the next day or tomorrow" and they went off scribbled what they had handy onto pieces of paper and you know you can tell that these drawings kind of look loose and a little bit rough maybe not as useful for design purposes If you're interested in this stuff, I will point you to one really useful reference, that is worth getting Its a classic book and also around the same time as 1879 Jule Boulgren, created a beautiful compendium of Islmaic star patterns I think it originally came as a booklet with a bunch of seperate plates Nowadays, it is very easy to get because there is a dover edition of just the plates and its really cheap so you can just get page after page of these abstract diagrams of these line drawings that represent patterns that he found around the world Now, again, addressing the mystery of how to construct these patterns, you can see faint, there are faint lines and circles in one corner of each pattern that try to give you some hint of how to construct it I wouldn't take those two literally. its not clear that they mean anything apart from "here's the receipe that Bourguen came up with after staring at the original patterns for long enough" he came up with some technique that allowed him to do a reasonably good reproduction so, you know, whether or not that corresponds, to what was done historically, is much more debatable Probably it doesn't That doesn't mean you couldn't potentially use these if you wanted to recreate these patterns But there is no system to it I will say more about that in a second Here's one more Of course, Islamic star patterns like these are being heavily practiced today, people make them all the time In fact, earlier this year when I was in London, I took a course by an organization that calls itself art of Islamic Patterns which is a couple of people who graduated A couple of artists, who graduated from Princess schoo, the Princess school of traditional arts in London and they now teach the construction of star patterns and floral design and they got to see how an artist views the construction of star patterns and they view it very much in a traditional way They want to draw these patterns by hand with a pencil That means going way back and picking out the original tools that were pretty sure artists would've used back in history. This, at least, we have a pretty good idea about because we know that the Islamic world were the people who were most versed in geometry in that era. This is, from my understanding, an Arabic translation of euclid's elements There is an interesting history here of course Euclid's elements were created around three or four hundred B.C. written in Greek, there were early translations of the elements into latin in the first millenium but they were basically all lost and forgotten by Western Europe around 1000 and its really through the Arabic translations that euclid was reimported into the West so we can be pretty sure that Arabic scholars were the ones in places like Bagdad where the ones who were really well versed in geometry and by extension, the artists were practicing stars patterns would have had all of this knowledge at their disposal its fair to say that in order to construct a lot of the patterns they constructed they would have used the basic tools of euclid and geometry. Namely, the compass and straight edge. When I took this course earlier this year, I was taught how to construct star patterns with straight edge and I want to show you an example. Its not important that you memorize or learn or understand every single step of what I am going to show you its just to show you how much effort is involved and what a recipe for constructing a star pattern may look like. So this is one that I actually did when I was in this class. You start with a line and you identify two points and that sort of sets the scale for the whole pattern, I'll go through this pretty quickly. Draw two arcs like this. This is actually part of the construction. This is the very first theorem and euclid's elements because these three points give you an equilateral triangle that is not actually relevant here. I get to draw verticle bisector there and that gives me a vertical line that is perpendicular to the horizontal one. Then I draw a circle starting at that point where the radius is chosen to exactly meet the center. I draw another circle with the same radius up at this point and now something a little bit interesting. It turns out that I already got three points of what is going to be a regular pentagon maybe. These two points are going to be two points of a pentagon and this is going to be the peak of a pentagon I just have to identify two more points down here of the pentagon and I can do that with just a few more arcs. Now I've got the five points. Very interesting though is that, to note, this is actually not a regular pentagon. It looks pretty good. Its pretty close to regular pentagon. If you are willing to do a little bit of math, a little bit of trigonometry, you can measure this length and this length, and this length. You can measure this angle, and this angle and this angle, and compare them to what the lengths and angles ought to be for a pentagon and you can see that it is actually very close. Its an approximation of a regular pentagon to within a couple of parts and a thousand. I am showing this to you, I mean it's not directly relevant in the pattern that is eventually going to emerge. You can construct truly regular pentagons which a compass and straight edge. It's interesting the way that we can cut corners. In fact, the way that you would want to because the other technique for constructing pentagons would be a whole mess more circles than the lines and other dodads that I need to achieve actually regularity. Because this is so precise, this is close enough that if you were to drawing it by hand, you wouldn't be able to notice the discrepancy. Or put it another way, the error you introduce just by the fact that you are using real world tools on a physical piece of paper is greater than the mathematical error in this drawing. So maybe it doesn't matter. Now, if you're fairly masochistic, you may redo this entire process to draw a second pentagon upside down If you're a little bit more savvy, you may get out some tracing paper so that you could trace the pentagon you just drew twice. So we are actually extending euclid's elements which the edition of tracing paper which is something he did not account for. Few more lines, we are going to identify the rectangle that contains a hexagon I am going to add a circle, so I have these regularly spaced points around it and it turns out that these are all... there are ten points based around these circles. I draw a few more construction lines that are easily found as joining various points in this diagram. At this point, things are starting to get a little bit crazy but I can pull out a subset of these blue lines. And get that diagram. Now I can throw away all the construction lines and now I have a unit which, if I repeat in a grid, I get an actual pattern. Okay. So that's actually... I mean... I didn't show you all the steps. What does it mean to repeat this in a grid? Again, if you're smart, you're going to do this with tracing paper. I will construct one unit and then I will put tracing pattern on top of it and then I will slide it along and trace out multiple versions, multiple copies of that unit. You know, I didn't repeat all the steps I needed to acheive a second pentagon and so on. I mean, this, there is nothing impossible about this, there is nothing that prevents anybody from going off and learning this technique and producing it. But boy, its hard. Its difficult manual labour. its easy to make mistakes. Which is not to say anything negative about the craft aspect of making Islamic art. It is highly worrthwhile. i mean I am incredibly grateful that I got a chance to practice it at that level. But as a mathematician and computer scientist, I look at this and I say this doesn't help me This does not help me understand Islamic star patterns and it doesn't help me make them efficient. If i had some technique, to help constructing a star pattern, and I build a tile and then I repeat it, and then I look at it. Mmm.... Its not quite what I wanted. Can you move that line a little bit over here and redo it? Well, that's a quite a painful thing to have to do. Whereas, If I can audit this to some extent with computer graphics then making mistakes much less painful. In fact, its pretty much completely painless. I will say more about that in a second. But the real problem with this, from my point of view, the thing that I respond to as a computer scientist, is that there is no enlightenment here. There's no insight in this recipe. This recipe is a hard coated sequence of instruction. That, if I do correctly, produces one designs and that's it. I can't adapt it to other situations, it doesn't teach me anything about any other Islamic star pattern. In some way, the recipe is as much a finished product of a design process as this picture. There was some underlying, a priori, set of ideas that inspire that recipe that I don't have access to. In some sense,the recipes that got passed down through history but in this pre digestive form that tells us nothing about how to actually design star patterns. Only how to draw them. Its a different thing. So the computer scientist says, show me a higher level abstraction. I want a back away from these hardcoated recipes and try to understand some kind of more mathematical abstract model that gives me a machine I can use to generate new star patterns on the fly. If I have a mathematical model, chances are I can encode it as the computer program. question in back: "according to the patterns?" Yeah well yeah, what's the method? What's the underlying idea that gave rise to all of these recipes. The recipes are beautiful things in of themselves. There are a lot of nice Islamic art where the contemporary art where you see the finished pattern but the artist has also chosen to give you a lot of the construction elements and thats a beauiful thing. ... that kind of scaffolding. But what I seek as a computer scientist is some underlying explanation for where these recipes came from. I want us to take a step back and understand if there's a model that I can use to drive the creation of new star patterns. Ideally, as a computer scientist, if I have that model, I can then find new degrees of freedom. New ways of vary the construction that I've just used in new directions that may be even traditional Islamic art wasn't able to explore before. Because we have tools now that weren't available five hundred or a thousand years ago or even one hundred years ago. We have much more sophisticated mathematics which is not any kind of insult to the brilliant work that was done a thouand years ago. We just know more now. We have beautiful theories mathematically for tilings, for tessellation and for understanding the symmetry of designs. Even simple things like trigonometry, we have calculators that can calculate the sin function and the cosin function to as many decimal places as we need. Which was a very, again, a very painful process evena few hundred years ago. So how can we use these modern tools and turn them into algorithms and give ourselves the power to create new star patterns? So here is my inspiration, there are a few different ways that people have proposed for constructing star patterns these days. Which are effectively reinventions of techniques, we don't really know what was done originally. But we can propose new ideas and hope that they give us some abilty to account for the structure for historical patterns, That's all I am asking for. I'm not going to make any claim that this is what woud have been done originally. So my inspiration came from Hankin who was part of the British archeological survey of India which would have been in the early 20th century or late 19th century. He saw a few examples of design that were done in plaster that had some extra construction line in them like the Topkopi scroll. From that, he came up with some ideas of how you might construct these patterns. One way to do it. So what he wrote in his paper was it is necessary to cover the surface to be decorated with a network of polygons in contact And thats why we call this the polygons in contact method, and then through the center of each side of each polygon lines are drawn. The lines cross each other cross each other like the letter x and they keep going into lines of similar origin. So you imagine at the centre of every edge, well what is effectively a tiling, a tessellation of plane. From the centre of every edge, you grow an x and you assume that you are growing a similar x from the centre of every other edge. And where two lines meet, where two arms of these x's meet, you cut them off and stop. You stop growing This is the key step in this construction technique, is the idea that I have these arms growing and I cut them off. I get one small element of my design and I do that everywhere. And this is something that can be automated very nicely. I mean, its very easy to write a computer program that kind of represents these x's and understands how to take the intersections of lines and cut off these drawings as they are happening. In fact, I want to make one more simplification that Hankin didn't. Which is, because we understand more now about tiling theory, than even we did in the 20s when he wrote this, I am going to restrict my attention to what's happening inside individual tiles. I don't really need to consider an x growing out of an edge between two tiles. I am going to look at what happens in each tile in isolation and I am going to grow, I guess a v, coming out of the centre of every edge. This is just one minor simplification but I'm going to look at what happens inside tiles one by one. So let me take these three shapes. I haven't exactly told you where I pulled these shapes. Out of thin air but You know, they exist, there they are and I'll say a little bit more about where I get shapes like these a little bit later. So this is a regular ten sided polygon. Regular decagon, this is kind of a bow tie hexagon, and this is more elongated convex hexagon. And I am going to draw my V so that the arms of the V make a 54 degree angle with each edge of each polygon. If I do that and I suitable cut off these arms when they meet other arms from other edge points, I get these motifs in the tiles. If you think about it a little bit, its pretty natural that if you grow a V out of every edge center, on a regular polygon, what jumps out as a result is going to be a star. So that's good! As long as I choose tilings that have lots of regular polygons in them, it looks like I'm going to get designs that have lots of stars. Cause I am going to stitch this together in a second. And the bow tie makes this design and the other hexagon makes this kind of design. Its just a natural byproduct of following that polygon contact technique. Now if I assemble the tiles in this arrangment, making an infinite periodic tiling and I copy those motifs into each tile, and then erase the tiling, I get the same design as I had before but I would claim that I have a little bit more insight into where the design came from now. I don't have a prepackaged recipe. I have a technique that's broken down into a couple of phases. First, I specify the tessellation I want to start with and then I have this fixed procedure to creating a motif for every element of that tessellation. That depends on one continuous variable which is the angle you make here. So I've broken the problem down into two sub problems. I need to tell you where tessellation is coming from and I need to tell you how to choose the angles and construct the design as a result. I guess, um, I'm going to jump here and show you how this works live. Because why not right? "So one dominant polygon and two subordinate?" Yeah that's a good way to thinking about it. I mean, if the goal of constructing the star pattern is to get lots of stars, Then I am guessing you are going to wan to have a lot of regular polygons and so in some sense, the goal is to construct tilings that have like big regular polygons that fill most of the space and then you somehow fill in the extra space with regular bits. Now the shapes that I just showed you are a little bit more special than that. These three shapes are part of a little more specific system. It's not fair to say that this and this just kind of rose and we dealt with it. These shapes are really themselves inshrined in the history of Islamic design as part of gilroy tiles I think I've pronounced that some what in the right neighbourhood. GIRIH in english which my understanding is the word for 'knot,' So you know in general, the idea is yeah I am going to try continue constructing the tiles by adding a lot of big regulat polygons. And then I will figure out a way to fill in the rest of the space with other stuff. I will say a little bit more about that but let me give a live demo because the point is I've reduced the problem down to this kind of formulatic system that I can encode in a computer program and that lets me explore a space of designs in a really quick and efficient way. There, is uh, its actually a little bit broken, but there you go. There's the tiling i just showed you although I guess its rotated? No I guess that's the right orientation I had. Let me just, turn that down, now I am going to turn this knob down Some of these of these knobs are clearly for demonstration purposes There we go. So there's the tiling right there and large. Now, what happens here that's fun is this actually will simulate the process of growing those x's or v's if you want to think about them that way. And I will grow them until they meet other V's and they just stop. I have precalculated how far they have to go before they stop and so I let them grow until exactly that point. And if i erase the tiling, then yes you get an interesting design. This is exactly the same design I had before I have to press this magic checkbox which Um, allows the lines to pass a little bit further into large regular polygons. In a way that I am not really going to go into. The other thing that is going on here is that I think this is at an angle of 45 degrees so I do have to I do have to adjust the angle to get closer to fifty four degrees and I realize as I am doing this that I can't see what I have got. That;s fiftyone degrees. It should go up in increments, there's fifty four degrees. Okay. So there's the design we started with and of course the nice thing about this is that you can't do by hand First example, that you can't do by hand, is this... ladadala, (laughter) No my house is really big I need a lot of this. Its trivial to make as much of this as you want and this is one of the real powers of computer science that we don't have with handmade art. That repetition is free. The computer is really good at stupid, mindless repetition and so I can break the task of design down into the interesting creative human part of coming up with these tilings and choosing angles and the boring repetition part which is copying of the design. Even if I wanted to eventually take this and use it as a guide for drawing or painting something by hand, this still lets me explore the space of possibilities much more efficiently than I can do just by drawing it by hand. And of course, the other that I have that I don't have otherwise are these beautiful continuous degrees of freedom. I can explore a whole range of star patterns in a way I would have to produce a hand drawing for everyone of these. just to see what it would look like. Of course, you know, once I have done that, I can plug in any other tiling I want. So I said something about where do these tilings come from. Well, there's a lot of possible sources of the tilings that are used in Islamic art. Some tilings have just been known since prehistory, I mean, the three obvious one are hexagons and uh squares and triangles And these do yield reasonable Islamic star patterns as well. Let me bring the tiling in a little bit there right? So simple exact tiling does produce a range of well known Islamic star patterns. if I set that to be, uh, sixty degrees, which again i can't quite see. Then I will get this classic design of six pointed stars surrounded by hexagons. Yeah go ahead. "I mean like we have heard like golden ratio in contrary, and like whatever, you had 5054 degree 45 degrees, and some how it wasn't as pleasing as the 54 degree" Yeah yeah, well was that related to the 54, golden degree ratio, probably not but 54 degree angle is related to the geometry of the decagon. that was used as one of the elements in there. Like if you look at, so why was it 54 degrees? This is a good question. What if I told you that this angle here is already 54 degrees? These shapes, all of the angles, in these tiles are multiples of 18 degrees. So in some sense, it is natural that the angle I choose here is also a multiple of 18 degrees somehow related to geometry of the tiles. Yeah, if you look at what the people write about these techniques, you see that they kind of say in the polygons and the contact world, they say oh yeah, for this kind of tiling you need this angle or this angle or this angle and the other range doesn't matter. You don't care. And they don't say why but the answer is its related to the tiles themselves, its related to the geometry of the tiles. The other thing you can do in a system like this which is really lovely is of course, explore different styles of decoration. So, of course, these simple line drawings are scaffolding that you should build upon with artistic interpretation. So, a very typical that you want to do is render these lines as interlacement and you know again, with computer graphics algorithms its very easy to do that. You just, there's this whole theorem that says if I have, well, the first thing you do is this: There is a theorem that says that if I can colour the regions in my pattern with only two colours, such as adjacent regions never have the same colour, which based on the nature of the polygons and contact technique, will always work. It just happens to work out that way. Then indeed, you can always also interlace the bands in such a way that every band passes on alternately over and under the other bands it encounters. So that turns out to be kind of nice. You can take this further, if you want, I don't know if this is going to work. See, here is a slightly more complicated drawing style, in which, interlacements are themselves, kind of thickened and realized as a pure piece of geometry. the nice thing about this is that if you actually cut these holes out of say, a piece of wood, then the entire thing would hold together. You'll have, like uh, a kind of screen. I'll show some examples of that. Q: Does the theorem you mentioned have a name? Which one? Q: the one you just mentioned. Oh! Um, does it have a name? I mean, I don't know. I first saw that theorem in a study book for the putnum exam. It's used as an example of an inductive proof. The fact that, these kind of designs can always be two coloured. I don't think there is a specific theorem that says if its two colour it can be interlaced. But I can suggest where you might look to find such a theorem stated. So if you ask me later i can try to find something. Again, I am trying to gloss over a lot of the really deep mathematical details because I don't want this to be a math talk necessarily. What else do I want to show you here? I mean there are a bunch of other types of other tilings of course. Here's one more just for fun. That's closely related to the other one that I started except there are no convex hexagons. Its just decagons and bowties. oh I did want to show one thing. Let me go back to these: I talked about the three regular tilings. triangles, squares and hexagons. There are also the next level of difficulty of tilings historically are the so called, archmedian tilings where every tile is a regular polygon. But they're not necessarily all the same regular polygons. And these have a long history as well, um, so I can make a design out of that. I'm going to go down to about 45 degrees. Now what happens is interesting at 45 degrees, is you can also move the starting points of those x's away from the centres of the edges. As I do that, what happens is I get little squares around the contact point of all those edges. You get sometimes, what are called, two point patterns. This is whole interesting range of design within itself. These are especially nice interlaced because It almost seems like a kind of chain mail. what you get is a whole bunch of small isolated rings that all interlace with each other. And of course, the main benefit here is how easy and efficiently I can explore a wide space of possible designs. Parameterized over the choice of tiling and angle of the two point patterns and so on. I guess you can parameterize over, I guess it could have a particular generalization. The next step I guess would be a matrix or a language to come? Yeah that would be cool too. i mean it would be possible to kind of invent some of Islamic star pattern programming language. Or do you mean more geometric design more generally. Q: yeah kind of like a more language type. It would be kind of cool. I thought about that a little bit on and off. In the end, I kind of prefer having a pretty user interface with knobs that I can turn. But the programmer in me does see the value in trying to abstract this into some kind of programmatic system. Q: You mentioned that a lot of these patterns were based on works, books and i am thinking if you ever move beyond a non, equivalent code. Not as far, very limited, I think there is one spherical design on sored pommel. Let me get to that, if I do have time I will get to non euclidian patterns. So I am just going to have to race a little bit too. Going back and looking at the topkopi scroll now maybe the polygons in contact technique may help us understand what is happening a little bit here. You can see that maybe these dotted red lines are the polygons in contact and indeed, its probably the case that these lines are forming a 54 degree angle with all of the tile edges. So you know, that is a little piece of historical evidence for the polygons in contact technique but again, let me emphasize as a computer scientist trying to create new Islamic star patterns. It is not crucial to me that this is historical technique. There are people who are very interested to know if this what was done five hundred years ago. I am interested in knowing, can I create things that are recognizably Islamic star patterns? I want to innovate and create new ideas that are in that space and that aesthetic. Where do the tilings come from? I already said this. you, uh the three regular tilings are very well known, you go back to at least thousands of years. There are other tilings that are relevant to Islamic star pattern designs. The use of archimedean tiling but it kind of looks like it. It is all made out of regular polygons. And here are two that look a little bit like the one I showed you, they have regular polygons and bowties together. Part of the research I did in Islamic star pattern design is to try to understand where these tilings can from and I developed a system for constructing these kinds of tilings based on the wallpaper group. Based on the plainer symmetry group and I am not going to talk about the details behind that but immediately it gives you access to these tilings which are a little bit weird and irregular because of these bowties. but no longer off limits. I can get access to them. I can account for their structure mathematically. Another important element in the design of star patterns are these rosettes. Often, you will find that a central star is surrounded by a layer of these barrel shaped hexagons. And we do want to account for that as well, now its possible to give an explicit compass and straight edged construction for the geometry of a rosette inscribed in a regular polygon and there are nice old papers on how to do this. The key is to, its actually fairly easy, the key is to find this point s right here. Everything else the rosette will follow from this magic point. And that adapts nicely to polygons with different number of sides and so on. But I also explored an interesting variation on that in which I offer a way of transform this tiling into a new tiling in which the rosettes arrives naturally via the polygons and contact techniques. So it tries to do everything with this one algorithmic construction and if necessary I will produce a new tiling that allows that construction to do what you want. So this is, I kind of like this, because it takes these mysterious rosette objects. And shows where they came from and shows why they might have risen in the first place. so I have these, uh, in this tab here, I should have some tilings that do this. Here's like ten RD which you can see, whoa, that's pretty complicated. I am going to try magnifying that back. there we go. If I go up to uh 72 degrees or so, you can see that these rosettes are just emerging as a byproduct of the structure of this tiling. Rather than needing to be explicitly constructed. Its kind of a nice innovation. So I have a system that has a given set of tilings that you start with and I account for the structure for a lot of these tilings and will be able to turn them into a range of interesting star patterns. The next question as a computer scientist with an interesting design is: what are you going to do with this? How can you make use of these to innovate and create new ideas in design? So I want to ake you on a tour and show you some of the stuff I have done over the years as variation. on star patterns. The first source of inspiration are metamorphoses. So Escher was very famous for creating these metamorphoses objects and I have spent a lot of time studying Escher. So this is something of interest to me. Escher was also heavily inspired by Islamic art. Escher travelled from the Netherlands twice to the Alhambra and he and his wife sketched a lot of these mathematical designs there. And he loved the beauty of abstract mathematical patterns except that he found them too abstract. He said, where are the birds and fish and lizards and so on? You know, Islamic geometric art tends to shy away from figurative representation , from animal and human forms but that was no good for Escher. So he took a lot of what he saw in places of what he saw in the Alhambra and developed them into real world patterns. Incredible stuff. Now William Huff, who was an architecture and design professor in the 20th century was also heavily inspired by Escher. He created this style of parquet deformations that he used to give as an exercise to students in his class. to draw these by hamd. It is a very challenging exercise. But I also , I love these, these are wonderful story telling, almost narrative flow in one direction as a tiling evolves and changes and undergoes metamorphosis so it was natural for me to ask whether Islamic star patterns could be subject to a similar kind of metamorphosis. In fact, its not even really that hard to do, I showed you that I had this continuous parameter that I could vary, this angle that I could make with the edges in the tiling so why don't I vary that angle within the scope of one design and you can see that I started to do that here. I can make the angle of contact be varying spacially in one piece of design. So you get these nice long linear designs that evolve, that are still reconigzably Islamic star patterns. So i call these Islamic parquet deformations and they're nice because I think they do live comfortably in the tradition of Islamic geometric patterns but there is no way people wanted to execute these historically because every single tile is different or at least, every slice through this tiling is made of differently shaped tiles. So there is no way you're going to hire tile cutters where every single one of these are unique. First of all, the labour would be immense and second of all, the chance of errors would be too great. Its just not going to fit together when you are done. But these days its not a problem. Another interesting, new innovation that has been applied me and others to Islamic star patterns is the use of apriotic tilings. So I am not going to go into the math behind all of this tiling. But they are very modern and they only really existed for about fifty years or so. That we've understood that they are possible. A apriotic tiling is one that doesn't repeat in any direction os its got this kind of rough irregular appearance but still a lot of structure. This is one of the most famous. This is one of the Penrose tilings. Now it turns out, its possible to put little fragments of Islamic star patterns in each of these rhombuses that join together to form this overall design. So let me take this 72 degree rhombus. I can take the same decagons that I had earlier and I notice that if I put a decagon at each of these verticies, they exactly meet each other and what's leftover is one of those bowties. Ha! Perfect. I know how to fill those up. Thats easy. I put these things in the decagons, I put this motif in the bowties and I am done. I cut it off to, you know, there's the design that I need inside one rhombus. I also need to do something for the 36 degree rhombus. Life is not quite that good. Because the decagons overlap each other when I put them down in the vertices of the rhombus. So I have to clean it up somehow and its a little bit adhawke but here's one possible solution. I get rid of a little bit of the extra complexity here and I end up with this simple design which I compromised and I have sort of, semi incomplete stars. And i get this motif. But now what's great about this is that i can join these rhombuses together and get what would in theory, fill the plain with a non repeating aperiodic, Islamic star pattern. There are other people who have looked into this topic. One recent one is by 2007 that got a lot of press was an article in science and the journal of science by Peter Lu and Paul Steinharot who are actually physicists. Who talked a little bit about the relationship between these aperiodic tilings and tradiional Islamic star patterns. So this is the Spandrol from the dar pleplum mosque in Isfahan. They claimed that the self similar stucture of this pattern suggests in some way that the artist that the created it some hundred years ago, had some understanding of aperiodic tilings and that is a very bold claim and I think its controversial. It caused a lot of negative response in a community that studied these as Islamic star patterns. I don't think they provided enough evidence to convince people that that is actually true. Its quite easy to discover this similarity without being specifically aware of periodicity which is a much richer, more contemporary mathematical topic. Nevertheless, we have these beautiful designs that geometrically seem to belong in a similar class. Ok so now, getting to what you were asking before. Can we adapt these designs to noneuclidean geometry. This is one of those topics that I studied as part of my pHD. So we have these fixed set of ways that tiles can fit together in the plain. If all I want to use is squares and only squares well I have to have four around every point where they meet. I can't have three or five. If its a 90 degree angle, I don't get to change it. But I can change it as long as I am willing to move off the flat plain. So for example, I can think about, what are the regular tilings of this sphere instead of the regular tilings of the plain. If you are willing to adapt to what you think of as a tiling, to a round surface of a sphere then you can make the legitimate claim that five regular tilings of the sphere and they basically correspond to five platonic solids. Tetrahedron, cube, octahedron, dodecahedron, icosehedran. In fact, if you look at theaxioms of euclidean geometry, we can't get all the way into it but if you look at the way euclidean geometry is formed, a lot of it works just fine on the sphere, you have to jigger it a little bit, you know, turn this and adjust this and polish this up but once you've done that,you can actually do most of geometry as you could conceive it all on these noneuclidean surfaces and that means that a lot of the techniques that I developed that worked on the euclidean plain also work in non euclidean plain geometry. They work on this sphere so if I can turn this tiling by hexagons into this thing. regular polygons and bowties. These are twelve and nine sided polygons and bowities I can turn this tiling by pentagons into this with decagons not notagons (nine sided) and bow ties. And the polygons in contact technique will work just fine. So I can take an originally plainer design and transfer it into a truly spherical design. The design is actually wrapped cleanly and perfectly around a sphere. That's just one half of the noneuclidean geometry though and in some sense, the more famous form of non euclidean geometry the true revolution in the history of silence was the understanding of hyperbolic geometry which was invented by a group of people arguably like lobachevsky and goust and a few others back in the 1850s. Escher was heavily inspired by hyperbolic geometry. He saw it as a way to capture an entire infinity and entire world of finite space inside of a canvas. There nothing left here to be represented every fish in an infinite plain has been mapped down in some hyperblic way to fit in this disk. And sure enough, all of the same techniques that i talked about also work in hyperbolic geometry. so i can take this pattern with 12 and 9 sided polygons and get this one that has 14 and 9 sided polygons. I kind of, to get 14, I need to make more space everywhere. In some sense thats why the hyperbolic plain gives you, I can transfer euclidean star patterns into hyperbolic plains as well. So here are a couple of finished designs with it coloured in and polished up and made to look real pretty. Its nice to group them together so that they are somehow thematically related so they have very similar structures but mapped into different universes. And one more with interlacements. so you can see this is made out of a lot of regular hexagons this one is made out of a lot of regular pentagons that interlace around each other. and this one is made out of one, two three, four, five, six, seven gons that wind around each other. There are a few other transformations on this plain that I don't have time to talk about. So you can pass your design through funny functions of the complex plain into itself. and get these interesting conformal mappings that you could apply to Islamic star patterns. They can applied to anything else as well of course. Here's someone else who is affiliated with the bridges conference that Zahala mentioned at the start who has done these nice multiscale Islamic scale star patterns moreorless by hand by discover ways that rosettes of different sizes could be made compatible with their neighbour. This is a very loose, much more free form design based on gira tiles that fit together to form a nice multi scale pattern. Joe Batolomew has a bunch of examples like this. One more kind of arguably, noneuclidean geometry but more free form would be Riemannian geometry where curvature could change everywhere on an object You can talk about what is the geometry on the surface of a rabbit instead of the very well behaved regular sphere. and hyperbolic plain. And it turns out there are nice algorithms in computer graphics for decomposing something like a bunny. into a bunch of roughly square regions. Not exactly square but close enough. If I find some symmetric patterns made out of square motifs, I can map those squares into the squares of this design. And get an arbitrary surface covered with an ornamental pattern. which of course you can then apply to Islamic star patterns. i don't think there's, I wouldn't claim there is some natural affinity, like there is not any reason why this has to be bunnies. The bunny in case you don't know is kind of a cannautical object in computer programming that can be used as a test case for anything. and the reason I am using a bunny is because that is the model I had available suitably processed so that I could map these designs. How I would love to be able to apply this technique to any model but I don't have the technology to process other models in the same way so i am still waiting to find somebody to give me that because it is actually my contributions, the fact that I tried to map star patterns in here is kind of trivial. compared to this step, the thing that actually found those squares in the first place. That's an immensely complicated algorithm and then i used it to do, you know, fun Islamic star bunnies. which kind of sounds like a science fiction story, that would be great (laughter) Final thing, mostly final thing, is just to show you a whole bunch of applications of this stuff How you can use this, the next great thing once you have complete computer implementations of star patterns is that there are so many wonderful technologies these days that the computer can attach you to create real world objects. So the most obvious one is a printer. Maybe a large format printer. This is actually not exactly my work. I was contacted by a furniture maker in the USA who saw my designs online and circular table for a client and he said it would be really great if we could make this be the table top so i sent him a super high res version of it and printed it out and sealed it in vinyl. then made this beautiful Islamic star table. I feel like, you know, you should have like a giant vase here and a big mug here and you need progressively smaller glasses. and stuff. (laughter) you put down in each of the circles. Laser cutting is really popular these days. I think there are a few laser cutters and various places on campus. So very simple, I mean it really is just press a button and go. Its not longer just printing something on paper. This is just a little square of wood that was cut with a star pattern. If you can define geometry of the star pattern programmatically You can fabricate it out of flat materials. This is a slightly more ambitious version of the previous one. This is a laser cut pattern in chrome plated steal I think it is. They've messed up the pattern a little bit. I provided the geometry and they had to make some modifications as a concession to the practicality of cutting this. but this is uh, I hooked up with a company in London that does custom drainage so this is a drain on the floor. I think this is in fact, the drain in a foot washing station in a mosque. The water is coming down from up here into the sink into this drain. It's kind of a nice idea and hopefully we can get more Islamic star pattern drains. Now they are so easy to design, we just need people to manufacture them. It is kind of a boutique item, it does, this type of custom manufacturing is more expensive then mass production but for something like an ornamental pattern in a mosque. maybe that is okay. and if you compare the cost of manufacturing this versus the cost of laying down thousands of tiny glazed tiles maybe, over like, you know like, the alhambra was built over decades. maybe it doesn't seem so bad anymore. Here's something I did earlier this year. This is a marquetry. The pieces of wood are cut with a laser and then manually glued together. You know, the thing to explore here are the deviations from perfect geometry. So here I was inspired by glitch art where you take jpeg images and randomly change bits of them until you get something where it is all distorted and broken, i wanted to see what that might look like geometrically applied to an Islamic star pattern. Here is another one where I took one of those spherical designs, projected it flat and cut the pieces out of wood. So its like a template if you look at it from the right angle, it kind of jumps out at you. Although it is very regular pattern, when you executed it in wood, every piece really is unique in projection. On the sphere they are all the same, but in projection they are all different. Here's a very old project using a computer controlled cutting bit to drill out the pattern in a block of linoleum. from which you can then do linoleum block printing. Its a nice marriage of the digital technology with hand crafting. As is the marketry You need the hand made element but you get the boost of the technology. A few more examples, this is cut out of paper. It is very thinly cut out of paper. This is not cut with a laser but with a computer controlled knife. which is a little table top thin and I recommend because they are really cheap So if you want to explore, this kind of design, it is only a couple hundred dollars to get your own knife cutter. And then you can play with this kind of paper design. The one on the right is much more heavy duty, its cut of synthetic marble or something like that. Now the spherical designs you obviously can't cut out flat material but it is where you can start with 3D printing and I have been doing a lot of 3D printing in the last few years and one natural thing to apply that to is these spherical Islamic star patterns. This is about three inches or four inches big. made out of plastic. this is one is actually an inch across, you can tell by the fingers, and that ones silver. So this way the silver works is that they print 3D in wax and then gloss wax casting from that wax positive. A couple more in plastic, if anybody wants to look after I am done in two minutes, I did bring the black one along So You can have a look. SO theres the black one, its only four inches across. Q: What about in gold? Shapeways will let you do 3D printing in gold, its not cheap, but maybe someday I will try gold. Most of the examples are in plastic. Here's, haha you know, have to do the bunny. So there is a bunny with the islamic star pattern mapped onto it in 3D printing. Okay so last little bit is where I want to go with this stuff what's next? There other styles of Islamic star patterns design that I haven't even touched on. The style I have talked about is the style that is much more popular in Persia. In Iran for example. But if you go to Morocco, you see a very different style in recognizably Islamic art emerge. So these are all fountains in Fez which hopefully I get to go to next spring. Se them first hand, these fountains are all executed in Zeliched tiles, glazed terracottic tiles, thousands and thousands of tiny tiles What you typically have that you don't have in the designs I showed you are a large many pointed central star with a design radiating out. That central star can get really big because there are some designs that typically practiced in the world about as one hundred points, actually 96 points. But I think they refer to it as a hundred. It is again made of all these very fine glazed tiles. The tiles themselves are pretty amazing. I don't know if you've seen the way these are cut but they fabricate sheets of tile and then cut individual tiles. out of the finished baked sheets. with fine axes. So they will scrape the designs into tiles and then chop off the bits they do not want until they get exactly this final shape as a result Incredibly hard. The really skilled craftsman can only make a couple hundred of these tiles a day. A fountain might require tens of thousands of them. I haven't yet developed the algorithms to produce these automatically but i have drawn some by hand. Just to see whats involved, what the challenges are. And its kind of interesting because you see that there are different zones where different things happen so a lot of this stuff is pretty easy and based on well known tile shapes. And the central rosette is pretty straight forward because it is perfectly symmetrical. Its this transition zone that gets hard where you have to reconcile this 24 fold symmetry with the eights and sixteen fold symmetry. of whats going on around it. And so, some of these tiles are compromises, they are not perfect, they are distorted or slightly irregular and thats something that computer science is not very good at whenever you need something that is not quite right, you have to tell the computer a lot of how that might work. You have to explain the nature of the compromise of the computer so it can somehow balance all the other opposing forces. Trying to make that shape be what it is. So that is a work in progress. Here is one more example of that kind of design. So I took another twenty four fold zelesh pattern and turned it into a laptop skin. this actually has a funny story associated with it. My pHD student who was working on an unrelated topic wanted to create a research poster that included me. So she said well you need to give me a design to put on your shirt in the poster so I gave her the laptop skin design so that its very clear that it is really me. And then the poster won an award at the conference where it was presented so I had to somehow respond to this. I responded by having the design printed on a real tshirt (laughter) There i am. One more I want to say is that some of these beautiful patterns seem to incorporate multiple scales at once. If you look at this pattern, you see lots of small elements that make it up. These are traditional zilead shapes. But you can also pick out some of the elements, colour them in black and then they seem to form a similar design at a larger scale. Maybe that can repeated again and again (etc) There is this beautiful notion of self similar multi scale designs as you can see here and in fact see in the workd of Lu and Steinhowart with these black lines. I think there are a lot of wonderful ideas that can be explored in how individual tile shapes can themselves be seen as composed. of smaller copies of themselves in a process than can repeat recursively down to infinity to make beautiful designs. So that is my crazy, brief tour, through some of the work i have done with Islamic star patterns. If you want to reach me or ask me about the map or computer science, the art, to the best of my knowledge, I am easy to find by email, web twitter if you are into that and I am happy to answer questions. (clapping) I'll pass this around if anyone wants to look at it here. Q: I think the last two slides, are they any examples of recursive designs, it doesn't seem like they, the artist discovered frontal patterns, you know... Yes and no I mean Like this to me looks like stars within stars but not quite at that fractal. Its not the same. You know when you say a star within a star within a star (etc) I really think of something more like that where you have things of all level of detail all interacting within the canvas. There certainly are historical examples where things are happening on multiple scales as once. In fact, Jay Bonar says there is one with three level all together in one design. There should be more. There is some of these and I don't know what they were thinking when they designed like I don't think they understood aperodic tilings but it would be interesting to know if they had some sense of infinity of the implied infinity that you can get from repeating this process. Q: maybe its rare because it is harder to make it on a large scale? Thats an interesting question. I've debated for a long time in my head just how hard this is like if I created an arbitrary arrangement of these small tiles. Can I always pick out a larger scale design? I sort of can, maybe you know these small tiles fit together in a limited number of ways and you will always be able to trace out these courser paths. Its not too hard to speculate that the typical way you may put these tiles together naturally yield this self similarity in a way that like some artisan 500 years ago thought, "yeah that is kind of cool." But not thought about the further mathematical implications. Something that needs to be explored more for sure. Q: what software are you using? Okay that's a good question. Well, no so okay, I mean the software that I am using: this thing is my own software that I wrote. its all written in java which obviously, as a programming language, java is getting harder and harder to actually use to maintain. This is a total nerd, computer, software engineer thing but java is falling out of favour in the world as far as I can tell. So I will stop using it. Yeah its true the newest version of java is a nice language but its not very well maintained anymore. the problem is that jave belonged to Sun which was bought by Oracle which is now part of hp. Anyway, yeah, there is a whole seperate question of software engineering issue. i would like to rewrite this software. I would love to make this like an Ipad app So everybody can use it. There's very limited availability of apps related to Islamic star patterns. There's one that came out recently and its really not very good unfortunately. I think there's a need there. If you were interested, I didn't get around to saying this. You can try my software online. Go to my homepage and you know what's easier? Its called tap rats. So if you just do a web search on tap rats. I'm sure you'll find it. So there's mine, my implementation of tap rats. There;s an online app like version you can try. There's supposed to be a downloadable application version. Its getting harder and harder like I said. If you try an applete, its even odds that your browser will tell you. " I don't want to run that for you." It could be dangerous. Browsers are getting fussy about java. But you can also go to this version. Someone took my older sourcecode and made their own applete called taprats. Anyway do a websearch on this and you will find it. If you are interested. "So that will give you those designs for this? Its not exactly the program you see on the screen there but its very similar. Eventually, watch this channel and their will be wonderful ipad apps you can used. Fingers crossed. "So the topkapi scrolls, do you know what the problem is with that?' I can't remember exactly but its about from the 16th century, its old. Whereas the Mursaq scrolls, nobody really knows. It kind of looks like they were done last Tuesday right? So the topkapi scrolls, they are much more like a historical document that people are serious about. Where Morzak bar, its not on display on the vna but its not as clear how historical. Its also more architectural than patterned based. The topkapi scroll, there is a wonderful, incredible book that is both a presentation and a deep analysis of everything that is in it. Unfortunately, the book is out of print. If you want to buy a copy, they are easily hundreds and hundreds, if not thousands of dollars. Someday I will have a copy that I can show you. You mentioned that the culture has a lot to do with the art. So i'm picturing, if western europe compared to the renaissance, the culture, different shapes and notions based on past greek arts, but to our eyes now, but what about their perspective did they ever seem to find that perspective? You mean in Islamic art? that's interesting! I mean there are, to my knowledge, no, you wouldn't There's no real opportunity in Islamic star pattern other then geometric design to express perspective and you wou;ldn;t want to because if you look at how Islamic star patterns are used,its used architecturally, drawn on a wall, book Any explicit use of perspective would make it really unsettlting to look at it. You wouldn't want a wall covered in tiles that reecede into a distance. Some kind of comple. There are other traditions in Islamic art. you know we talked about islamic art as all abstract and never figurative. Its not true. There is a tradition of minature painting. Which you see human figures. You would never see figurative paintings in a mosque but that does not mean people would not have portraits hung in their home. If you look at a lot of figurative paintings. You don't see a lot of perspective. So I don't know... it's an interesting question. I don't know what the relationship is between classical Islamic art and the perspective. I suspect that it does not play an important role in the artistic tradition. You have a comment on that? "I think that from my understanding that the tradition and Islamic tradition comes from different ways of thinking about how to create art." So from renaissance and Michaelenglo, to create a picture that always ressembles real life. Thats kind of thinking is different from Islamic artists and people who do the tiles do not think about any kind of replication of life but starting with patterns, you see what comes out of it. So that's kind of difference creates different base of thinking. Even if you look at miniature paintings, you think of a story telling quality instead of trying to create what Michaelanglo did which was study of human details to create a drawing Can you make the controversial case that its like ancient egyptian? is that how that used it? I don't think that's right. Yeah (laughter) Thats interesting and I've never heard that either, I mean I think they said "oh yeah we got perspective, yeah sure!" No we are above it, we don't You would think there would be some evidence of the use of perspective due to the creation of renaissance But yeah, I liked that. Thanks for that comment. I mean, We did kind of math and geometry that they had. I assume that they could've considered at one point or two point perspective. Its just a matter of what I want to have and that kind of interpretation. That's a interesting, its a nice. It'd be fun to explore that more deeply. i would love to see, the one example where some Islamic scholar drew something in perspective. But I have never seen it. I haven't spent a lot of time with miniature paintings but they tend to be more symbolic. More like cartoons. Liek you said, narratives. "They are sort of spacial configurationsand a different way of thinking what is closer, is it down? Sort of like things that go up and down then really going in. I'm thinking of, yeah you mentioned the camera, photography, even in our generation, photography is kind of like, like a reaction against that abstract. Even though it kind of sounds like patterns, it kind of sounds like, its like abstract and avanat garde, away from a realistic picture. Like I'm thinking like I don't know maybe there is something possible we can analyze? Yeah and I think that is such a big difference so the innovation of photography basically took away what a lot of artists were trying. Or what they were capturing real life. SO they basically had to start over defining what a painting is maybe not making a mosque look like a real mosque and what can they do? That opens up this space for abstraction. Question there. I have a question: I see that most of the work that done by the Islamic art consists of five goals or seven goals which are all constructive goads,,,? Uh not seven or eleven, if you mean constructable in the formal mathematical sense, five is but eleven is not. Uh... yeah Eleven won't be contructable. Seven isn't either but as with the pentagon there are Islamic star patterns with seven pointed stars. Traditionally, those would have been drawn using clever approximation using seven gons which is great. "And ten gons as well?" Ten is constructable. If you can construct an end gon you can also construct a two end gon. That's not hard. "Did they have a method of constructing these end gons or ?" Well some of them would have been taken from euclid and it would have been done with a compass and straight edge, There's no, as far as I know there is no general method to constructing an end gon. Uh, likethat avoids using trigonometry. To construct an arbitrary regular end gon, you kind of need to know of size and cosizes and you need to compute them. Otherwise I'm not sure, itd be interesting, there would be some way to take seven sticks and put them in a circle... and sort of juggle them until they even out as much as possible. I'm not sure. Um, but five for sure they would have known how to draw a regular pentagon. I kind of don't know what they did irregular stuff for seven and eleven. I guess there is at least one example that has got eleven sided polygons in it. Whether they would have just eyeballed it until it looked good or... done some kind of approximation based on an understand of the math of what. I ... yeah maybe, someone somwhere maybe freehand drew a really good eleven gon and everyone from then on went to his house and traced it for like (laughter) fire. You know you have to perserve the eleven gon for all of history (laughter). Yeah? "So you called them Islamic star patterns and I'm thinking what's a distinguishing factor. If you go to office, they have star patterns on the table..." "Like a stream, five , six, seven, eight stars. so what's the distinguishing factor of Islamic star patterns." Recurring patterns? There's no one true definition of what makes an Islamic star pattern an Islamic star pattern and its not a private club. I mean its really just the historical collection of things that were, that are considered Islamic star patterns. If you look at like, books of Islamic art, they have a hard time and they struggle trying to define what Islamic art is And that's fair, as long as Islamic art is practiced more than a millenium, easily fourteen hundred years, um, all over a huge part of the world, with a huge part of, styles, and a lot of it was practiced by people who weren't even Muslim. Right? But were Jews and Christians living califit rule. So whats get to be called Islamic art at all? "But like characteristics like recurring patterns, I guess like self symmetry? I mean like, I can't remember what I wrote in my thesis. I mean like it would be unproductive trying to spend a lot of time coming up with... a formal, mathematical definition so I can test, does this have Islamic star pattern nature or doesn't it? Uh, its better to say that its just a pattern made out of a lot of regular stars and whatever else is needed to join things together. But you know, what makes something Islamic art is what people say it. Just as, with the rest of art. There are people practicing Islamic art today who produce things that are very very different from what I have been able to show you here. I mean I saw, uh, another thing that I saw at the Victorian Albert museum earlier this year was the gallery of the finalists of the gemiel 3 prize which is a an annual prize given out for contemporary Islamic art. You know, there's wonderful things going on in contemporary Islamic art that would not look anything like this. Zahail has done some incredibly nice installations that pay homage to the Islamic art but look nothing like it. I mean, laser cut palettes, right? I mean, seeing people, the work that you do is very important to contemporary artist because as a contemporary artists I may not be interested in... the craftsmanship of making something that tedious because partly because the biggest part of the work that I do is not craftsmanship. Its the concept behind... it, its the actual creation that comes from that cultural background, or comes from ideas that I have, so it actually works better for me sometiems if I take my hand away and le a laser printer to do an image, something that is easily done but then as an audience to come in to thinking of these ideas that are underlying rather than the objects as a precious art object that you spent hours and hours to make. "As the kind of artists that does these patterns, and to take that away from you, do you kind of lose something in that process if you don't spend the time and do these or something like this? "No I don't" Because it makes my work harder to come up with ideas that I can attach to rather then making these things. Its a different way of creating art. Right! and that's your personal expression of art. There are other artists who practice this who would undoubtedly think that something is lost when you think of the computer as opposed to by hand and that's great. I want there to be people who continue to practice these designs by hand because they will see things in the design and innovate in ways that I won't Maybe I can use some of those ideas and maybe I can't I just have to admire them. Other questions? Or should we. We are already running over I guess. Any other questions? Thank you everyone for coming. Thnaks for sticking around. (clapping) Thank you professor
Properties
 Decagonal numbers consistently alternate parity.