 Welcome, everyone. Thank you for joining us at this geometry session run by the King's Foundation School of Traditional Arts. In this session, we are going to be looking at a pattern from the asm palace and this pattern that we will be looking at is an eight fold design. And if anyone is interested in learning more about the patterns in the palace, we have. Recording of a lecture by by Dr Cynthia Finlayson, where she's telling about some of the history and context of the palace. Let me just orient you a little bit to the pattern that we will be looking at today. So if you have joined us for any other classes in the asm palace series, you will be familiar with these glimpses. Some of the courtyards within the palace complex. You can see the beautiful striped multicolored stone. You can see some of the. Lines of decorations and particularly the roundel decorations that are arching over the doorways and windows of the palace. We're going to be looking at some of these designs a little bit more closely. And the design that we're going to be looking at today is an eight fold design that is setting up a relationship between a central eight fold motif and a ring of octagons. So I have several examples of this kind of pattern that I'll be sharing with you and then we are going to zoom in on one particular example of this type of design. So if you look at this roundel here, this is a made in colored stone paste in set into the the brown stone here. The this particular roundel. If you look at the very center of it, I'm calling it an eight fold design because we can see that it has an eight fold structure to it. That central white area is surrounded by these light Octagons that have a little four pointed star within it and these octagons are separated by these longer bridging shapes that have little spikes on either end. And it's this longer bridging shape that you can also see over here symmetrically on the other side of the of the arch. I want you to keep an eye out for this bridge shape in the next slides that we look at. So here's the same the same design in a different location in the palace. In this case, the roundels are further out on the arch. We have a close up over here on the left. You can see the bridge shape and the octagonal motifs that separate it and create a nice proportion with the central eight pointed star. Now, if we have rosettes in the center instead of stars, the design changes ever so slightly. So here we can see the roundel is an octagon and it has partial rosettes on each corner surrounding a rosette in the center and they're separated. In this case, those bridge shapes are white and we have the octagons again. So the octagons are being used to determine the space between the rosette motifs. Now, this example is a little bit more quirky and interesting. That bridge shape has taken on a new form and it has a bend in it because an added ring of octagons has been added to the outer perimeter. The four of the corners in this case have partial rosettes and then the other four corners have this little adapted red shapes. This one I included because I thought how you should see this, how that bridge shape has been adapted for yet another context. You can see there's a rosette on one side. In this case, the bridging shape is almost a full ring around it. And you can see that it's helping connect all of these partial octagons with that rosette. You see, here's the motif. It's just a little tiny motif. It's been cropped in quite an interesting way to make this unique design at the keystone of one of these arches. Now, we're actually going to be looking, not at a roundel this time, we're going to be looking at this wooden painted panel, which is inside one of these window bays. And you can see it here is a face on photograph of it, and you can see how that bent bridge shape is here. And this bent shape is providing a transition from the central rosette to a corner rosette, and then all of those interstitial spaces are filled with octagons. I'd also like you to notice at this stage the orientation of all of these octagons. They're all standing on a point. So when I talk about a dynamic octagon, I am talking about an octagon that's oriented so that it's on a corner, as opposed to, over here on the left hand side, a static octagon, where the roundel is a hole is oriented on its flat side. And that's what we mean by static. For our pattern drawing purposes today, we're going to be working entirely with dynamic octagons. Now, I wanted to put these two decorative motifs from the palace next to each other, because they are basically the same design. You see the central rosette and you see four corner partial rosettes with the intermediate spaces filled with octagons. For the roundel context, they have changed the containing shape to an octagon, which means there are certain geometric compromises that are being made. But hopefully you can see that if you were to connect the center of this black rosette here across to the center of this one, the square containing that pattern is identical to the pattern that we see in this wooden ceiling. So there is this repetition of motifs. So all of the patterns that I've shown you are ones that are dealing with these slightly odd forms that are bridging the empty spaces between the octagons and whatever the main motif is. This kind of pattern shows up in another context that maybe some of you will also be familiar with. And this is the Big Men Bar in the Ibn Talun Mosque in Cairo. And you can see here in this case how those bridge shapes are creating links between the eight-fold rosettes and octagonal shapes. And I have a link here to a really great site. If you find this pattern interesting, you can go into great depth on Alan Adams' website, Drawing Islamic Geometric Designs. And he's got several tutorials that you can download to study a set of related patterns to what we're doing today. But let's go back to our pattern in question, and we are now ready to draw. So I have got a three-size paper here, and we are going to be drawing on it in landscape format. And this is to make sure that what I'm drawing fits on the camera for you. We'll start with a vertical line running through the middle of the page. I have a radius on my compass, now about 10 and a half centimeters for my North American friends. We're looking at about four inches. And our whole pattern is going to be inside the circle, and is actually going to be considerably smaller than what's inside the circle. So if you want to make your circle bigger, by all means do. So I know this is, oh, it is just barely on camera, that's great. So in order to develop this pattern, it's in a square. So that means that we need to have our initial circle divided into quarters. And in order to do that, I need to construct a line that is perpendicular to my vertical. And I am going to do that by keeping the same radius on my compass. And just, I'm going to add these arcs coming out of the circle, but I want them to intersect. Having done that, I now have my initial circle divided such that I've got four additional intersections. I'll use these four intersections as a way to step out my radius. I'm putting the point of my compass at the intersection, and I'm striking a pair of arcs off to the right of my drawing. You can see it's just about on the screen there. And I'll do the same on the left. And this pair of intersections is going to help me get my horizontal bicepter in this case. I should have had a longer ruler. My initial circle divided into four. The final thing I'm going to do before I pause and ask for people to give me a progress report is I'm going to finish off this to make little ears around my circle. So I'm taking that same radius, putting it on the horizontal intersection, and I am creating these little ears all the way around my circle. That will be useful at a later stage. Okay, so when you have your circle with these two sets of ears on it, give me a green check mark in the reactions. That looks like people are doing quite well with that. I'll give you a little bit more time. Okay, wonderful. The next thing that I'm going to do is I'm going to shift my circle division from four into eight. And by having these ears, I can use the tips of those ears to draw diagonal lines across the composition. I'm now going to draw a static square. And this is going to be where the diagonals cut my circle. And this static square is going to be the square that contains the entirety of the pattern. Okay, let me know with green check marks when you have your bounding square in place. I think they're looking good. I'm going to move on to the next step. So within the square, we're going to add a bunch of construction lines that will help us hold the pattern together and properly proportion the different elements in relation to one another. And all of these proportions that we are developing have to do with the square root of two, which is the relationship between the edge of a square and its diagonal. So if the edge of the square is one, then its diagonal is the square root of two. So already in looking at our drawing, we can see that we have a square. We have its diagonal. That's divided into smaller squares and their diagonals. So at different scales, this particular proportion is essential for understanding and mapping out this pattern. And this proportion is a very key to the eight fold family of designs. So in order to develop this relationship in a way that will help us find our pattern, we're going to be adding a series of diminishing squares. And every step that we are making here is derived from previous steps. So we've, because we've just drawn our bounding square, we can now take its midpoints. Draw a dynamic square. Sorry, Lisa. Can I ask you a question about a step back? Where did we get the central crosses on the horizontal line? That's the question. Is that referring to these arcs out here? Yeah. Yes, please. Okay. So what these were, these intersections out here helped me figure out where to put the horizontal line. So this is the same radius as my initial circle, which you can see here. I had already divided my initial circle by drawing these arcs here. And then I took that same compass measurement and put arcs here and here. So this line is actually extending out here. So the horizontal line comes because we found these intersection points to the right and left. We can't put the horizontal line in the right place until we have those. So hopefully that clears things up. So what I'm doing now is inside my pattern bounding square. I'm now drawing a dynamic square. So a square standing on its midpoint. Connecting the vertical and horizontal lines. And this particular square, the dynamic square is half the area of the large static square. And we know that because the lines that we use to make it are cutting each of these in half. So we're only using half of this square, half of this, half of this and half of this to create a dynamic square. Let's do one more square inside this one. So this is a static square. And in this case, we are taking the points where our diagonals crossed the dynamic square. And now the square that we're drawing is the same size as one quarter of our initial square. Now let me know what the green check marks when you have these in place. Okay, that's looking very good. Thank you. Now we're going to now add in some circles that will help us position some key elements of the design. And the circle that we want to put in first is this one right here. And it fits inside that central square. Now before I go ahead and put that circle in place. I'm just going to double check that the radius is correct. And I'm able to draw a circle that tangents on north, south, east and west. And this is going to be the circle that holds in our eight-fold rosette. Now if I go back to the pattern, the circle that we just drew holds in that central rosette. And this design also has four quarter rosettes, one on each corner, and they're the same size as the one in the center. So that means that we can keep the same radius and put circles on the corners of our square. And that can be full or partial circles. So I'll do a full one here. But if you can only fit a partial one on your drawing, that's also fine. Okay, we're going to now need to add some more structure lines to help us figure out where to place some other small circles. We're going to first divide our central rosette into 16. There are a few different ways that we could do it, but to keep the center of our drawing as uncluttered as possible. We are going to be drawing a square that's the same size as our bounding square. And we are going to be doing it dynamically like this. I'm not connecting north, south, east, and west, but I don't actually need the full edges of the squares. I just need to know where it divides the edge length of our pattern. So here I'm connecting this point over here on the west and the south of my drawing. I'm just going to, I'll draw just the parts of that dynamic square that are outside my pattern. But what I'm really interested in is that intersection right here and right here. So just move around the drawing. In this case, I'm just going to put those intersections. I'll put in a dashed line so you can again see where they come from. And then on the top of the drawing, I'll just put the intersections themselves. So let me know when you have these eight extra marks on your boundary. That's looking good, but I'll leave a little bit more time. So if we, if we look at these eight intersections that we found in our drawing, we can use these eight intersections to divide our circle, our central circle into 16. So if I line up my ruler by taking these opposite intersections, I've got here one here on the bottom and one here on the top and my ruler is going through the center of my pattern. I'm going to draw a line now before you follow along. Let me just show you where I'm starting and stopping this line. Still construction lines, not a pattern line yet. I'm adding this line so that it is extends beyond that central radius and is held inside this dynamic square. So I'll do four lines like this and go ahead and put these on your own drawing and Russia, if you could clear the feedback and then they can let me know when they've got these divisions in. So if you have these divisions and you look at your central circle and you can see that it is divided into 16 equal slices, and these lines are extending out until they hit that dynamic square. So give me a green check mark when you've got that. Good progress with the check marks. Good. Another few moments. Okay, wonderful. Now my central circle is divided into sixteenths, but we can use this information that we've just found in the center to help us also divide the corners into sixteenths. So if we think of the edge of the dynamic square as a mirror, we are reflecting these 16 fold divisions back into the center over here. I can line up my ruler. And when I line up my ruler, you notice that it is picking up the center of the corner, the intersection with the diagonal line and all the way across here it's picking up the eight full division on my initial circle, the horizontal line. So those are different ways to check the accuracy of this line. So every time that you're adding these sixteenths marks, make sure your ruler is lining up and picking up one, two, three points across the pattern. And as you can see here, we've completed this diamond shape that connects the corner rosette position to the central rosette position. We do that on all four corners. And when you have that on all four corners, let me know what the green check marks. So let me know when you reach this stage. And this stage means that the center and all four corners are divided into one sixteenths slices. One more moment. We're getting very close to being able to start working on the pattern. There are a few more little construction lines to add. Now you notice that as these lines cross here, we've got these places where we have three lines coming together. And we've got that, again, four places, north, south, east, west on our initial drawing. I'm going to be lining up these intersections across the drawing to add a few more little lines. So this one is connecting the left bottom one with the left top one. And I'm just drawing a line that goes from that point to the edge of the pattern. And I can do that to do another set of short lines parallel to that one. And I can do that horizontally as well. And the last thing I'm going to be doing with those intersections here is I'm going to close off the shape at the bottom. So what you can see here in this space is we have a double square or half of a square. And we know that because we've got, there's that root two relationship where we have the square shape and it's diagonal. They're waiting to be used to develop the pattern elements. So we need to close off those double squares or half squares on all four sides of the pattern. We'll be ready to pull out our compasses again. All right. So green check marks when you have this, those numbers are going out nice and fast, which is good. Okay. So we'll pull out our compasses again. And we're going to be doing smaller radius. So this becomes a little bit tricky when if you don't have a smaller compass, but we're looking to find the radius of a circle. That would fit in between these two spaces here. So checking before I draw any circles, I'm not just going to check it in one corner. I'm going to check it in all four corners and just delicately tune the radius until it is a good fit for all locations. I'm going to put in these circles that are tangenting the two big circles and all of these little circles that we're about to add, these are the ones that hold those floating octagons. So we're going to have a bunch of these in various places. So I'll go back to the screen share oriented us to it. We have just drawn the circles which contain these octagons, these dynamic octagons that are linking the rosettes. We need to find the locations for these other octagons on the outside. And in this case, the position of these other octagons is at the top of where those little double squares are. And it should just messle in perfectly so that it's tangenting the inside of our dynamic square. So we'll do those on all four directions. So now our central circle is surrounded by eight smaller circles that tangent it those points. Quite harmonious the way they all fit together like that. We have eight more of these little circles that we're going to draw. And the positions of these eight final circles come from where these vertical horizontal lines intersect are the boundary of our pattern. And you put this one in notice how it tangents both the corner circle and this little circle right here. And you can see already this weird little gap between them. That is sort of a curvy version of that bridge form that we are paying attention to in today's pattern. It's so beautiful and so satisfying the way they fit together so harmoniously. There are multiple ways that one can draw these patterns and arrive at the same results. And if you find teachers that take a different path to the same results, do learn from them and compare and contrast the methods. Each one is going to give you new and deeper insights into this wonderful world of geometry. So when you have this on your paper, give me a green check mark. Numbers are looking good. Another moment. If your circles are not tangenting perfectly, one thing to be very aware of in any sort of geometric drawing is that early mistakes sort of accumulate. And so, for example, if your initial horizontal and vertical lines are not perfectly perpendicular, things can go a little bit off in your drawing. This is just a matter of practice. If you are struggling with accuracy, there are a few things I recommend. Using a very hard pencil in your compass so that the line stays crisp and any straight lines I'm using normally at least a 3H so that my drawings stay crisp and unsmudged. The other thing is when placing your compasses, you need to take extra care to place the center of your compass. If the middle of your circle, if the origin of your circle is not in the correct location, everything about the circle is off. So, yeah, it is a matter of practice. So we're going to focus on completing the rosettes next. And in order to do this, what we need to do is add one little line that will help us find an important radius. What we're doing here is inside my central circle. If I take a line that is going as if it were going to be a dynamic square. And then I'm going to adjust it so that it's one sixteenth more. That's what I'm looking for. So do this as a dashed line. Some of you who've taken several of our geometry classes will understand why this line is so important. But what we are looking for is where this line intersects right there and also right here. And this is going to give us the radius for a new circle. And this is going to be a little bit larger than what we already had in our compass and put this circle in. Now, whatever we do at the center, we can take the measurements that we've discovered there and transfer them to the four corners. So I don't need to put this dashed line on the corners because I already have that measurement. I'm going to just go ahead and take that on my compass in each of the four corners. The circle appears to tangent the little ears. I don't know if that is an exact tangenting or just a coincidental close relationship. It would be interesting to check out the mathematics of that at some point. But what we now have in the center is what we, a governing circle, and I'm going to be able to use this circle to help me build my rosette. I'm going to do that now by connecting lines parallel to each other. Before I do that, let me just jump back to the original pattern so we have the rosette in our mind. So you can see that this rosette is made of pairs of parallel lines running across, running at 90 degrees and 45 degrees. So vertical, horizontal and diagonally. And so what we're going to do is to get our vertical lines. We're not using the central line, but we're just sliding over one sixteenth from that. And I'm just sketching these in. I am using a little bit of a darker pen at this stage, but it's not my final final lines yet. Actually, I'll just go straight across for this. So you can see that these sets of parallel lines are just their one sixteenth away from the vertical line. I'll do that one sixteenth away from the horizontal line and they're intersecting with that guide circle that we've put in. So I've got my vertical and horizontal and then now I can do my diagonals. Now, as I do my diagonals, notice that by connecting those two points, my ruler is also lining up with the guide circle and the one sixteenth divisions of the rosettes in the corners. So I'm going to add those lines in as well. Do that on the other diagonal. The corners, we are going to use pairs of the corners to help get the vertical and then the horizontal lines. So when you have all of these in all four corners in the center, let me know with the green checkmarks. Once you get very familiar with this pattern, and some of you will know this already, you don't have to draw this full line. You'll know where to start and stop it so that you just are going straight to the completed pedal shapes on the rosettes. So let me know when you're at this stage. I'm missing a line. I see a few green checkmarks coming up. I know that last little bit moved quick swiftly. So I'll give you plenty of time to finish that for a few more of you to report in before I move on because we're going to add some more construction lines. All right. The thing that to be aware of at the next step is we've got a lot of the lines of the rosettes. We haven't closed off the pedals. We could do it with the information we have on the pattern, but I'd like instead to focus on where the octagons are going to be. In order to draw an octagon, you need a circle that's divided into eight and none of our circles right now have that division completed into eight. But that is easy enough to do. So what we can do is extend lines that we already have or use points across the pattern to connect up and get those missing divisions. So for example, the ones that I'll focus on first are the four circles that bridge the gap between the rosettes. And this is very easy to divide them into eight just by extending the lines of the static square horizontally and vertically to get the design. So you can see now that this circle, it's divided into eight easy peasy. So let's make sure they all have that eight fold division in them. And we could have drawn all of these lines long from the beginning. But I wanted you to be able to see the why of each step. So these four are divided into eight. The next four that are tangenting the central circle. Again, very easy. What we can do is draw in diagonal line. Now what I've done is I've lined up my ruler so I'm connecting up the centers of these two circles and noticing that it's also in these, the ones on the outer boundary of the design also align there. So I'm going to put in these diagonal lines like so. And I'll do those. It's kind of like drawing a big dynamic square on the pattern. Now if I look at all of these circles, the last ones that need eight fold division are the little ones around the outer edge. And again, we can connect them with each other, the centers, and just add those missing slices. I'm aware that we're getting to the end of our scheduled time together. So I'm going to just going to move on if you're only able to focus on a quarter of your drawing at this stage. Make sure that all of the circles in that one quarter have been divided into eight and then I'll use those to finish the pattern. So if we look at any one of those little circles, we can see that because it's divided into eight, we can put a dynamic octagon inside it by connecting those lines. And you can also see there's these beautiful alignments that happen all the way across the pattern. So if I'm focusing on this little circle here, the one that bridges the two rosettes. If I line up like so, I've got one, two, three, four, five, six, seven, eight, nine different intersections that I'm lining my ruler up along. So I'm going to just focus it on this little one quarter of the drawing right now and just that one little bridging place right there. Just for now. So I've been able to pick up quite a lot of lines with that. I'll now work to close the octagon in the other direction. So again, keeping all my attention on this one quarter of the drawing, you can see the sides of that octagon beginning to come in. In this case, I'm picking up intersections at the edge of that one, two, three, four, five, six all the way across the drawing. So lots of beautiful connections and alignments. I'm still just focusing on this one octagon. So this line picks up a lot. So I've got one, two, three, four, five, six points again that I can use near that in the other direction. And to close off the last two sides of this octagon, we're picking up points here and they go across central rosette and out the other side of the drawing. So just by focusing on that one octagon, we have other lines that we are able to fill in all the way across the drawing. You can see that we have most of that bridging shape completed and we're just missing two lines that would help close off the shape. So those come from these little octagons here. And if I line up my ruler to complete this, it also lines up across the drawing there. This one should line up. It takes a little attention to make sure you're lining up the right points. It's actually this little bit here that has the least alignments of anything that we have worked with so far and there. So this is closing off this part of the design. Just shade in shapes. We've got our nice bridging shape. It's quite a strange shape, but due to the regularity of the rosettes and the octagons, it really feels like it belongs. This is part of the little four pointed star. And then if you shade in these kite shapes, you can see the pattern come together just as we see in the beautiful asm palace. Let's finish off this little corner here. You notice that we can't quite close off those kite shapes. We can do that. I'm coming in here and I'm connecting where the this edge of the kite hits the boundary line of the pattern on both sides. And it just comes across like so. So this is one quarter of the pattern. And that can be repeated all the way across. So I am going to quickly go in and fill out the rest of the pattern. So just keep working on your designs on your drawing and whatever you've done in the one quarter, just keep doing it around and look for those moments to extend your pattern lines all the way across. So I'm going to just pick up speed as I do this now. So I think this is quite a fun little pattern. It's a bit different from some of the other motifs from the palace that we've been drawing in this series of patterns that we've spent over two years picking out motifs from the palace. I thought this would be a fun adventure for us. Do keep an eye on our websites. We will have other courses in this series, which will come up in the coming months. The next set of patterns that we tackle from the palace are all going to be tenfold designs. So if you see them on the website, they'll be completely new. We won't have done them as part of the series before. So that's just done. And just a reminder that this particular pattern was intended to, or this particular drawing session was intended to complement the lecture that we had on the Asim Palace by Dr. Cynthia Thinlayson. If you want to know more about the palace, its history, its decoration, do look on the school's YouTube channel for that recording. And for those of you who are currently involved in that class, you will also get an email with a link to that in the near future if you haven't already. So look forward to seeing your completed drawings and paintings and other craft interpretations of this pattern on social media in the coming weeks. So don't forget to tag the school. You're very welcome to tag me as well. And really, thank you for joining us. We're very happy to have you with us. Hopefully everyone has been able to follow through with the drawing to be able to come up with a completed design that you'll be able to share with us. So thank you so much. And we are looking forward to seeing you on our other online classes with the school.