Draw and Paint Geometric Art Patterns with Voronoi Diagrams

1. Introduction: Voronoid diagrams are geometric patterns that frequently appear in nature on the coats of giraffes, in the motifs of leaves, and dragonfly wings. Their geometric structure is often an inspiration in design, architecture, and art. They feature in porcelain decoration and can be found on buildings such as the Water cube aquatic center in Beijing and the Gold Coast Art Center in Australia. Hi, I'm Diana, a mathematics teacher and a geometric artist. In this class, I will be guiding you step by step, how to draw your own Voronoi diagram art. I will teach you how to apply just three simple geometric concepts using a compass. So you can generate your own infinite variety of polygon tessellations, as well as circular designs. Then I will demonstrate a range of fun coloring techniques and texture explorations, such as water color layering, using salt, masking fluid, and even bubble wrap. This class is suitable for all levels, and it includes visual instructions, as well as ready, traceable templates. The skills learned in this class can be applied to any of your own future art. 2. Project & Materials: The project in this class is to create four designs based on a Voronoi diagram. Now Voronoi diagram is just a way of splitting the surface in an efficient way by finding regions which are closest to a given point. For example, any part of this region is closer to this point than any of the other points. That have huge applications in the real world. But of course, it creates really interesting patterns and designs, which is why we find it so often in nature as well as design and architecture The way to find where the lines go and create the actual polygons in the pattern is to find the shortest distance between the two points which splits it in half at a right angle. You're going to learn three simple techniques, how to split a line at a right angle in half, how to split an angle in half, and how to draw a perpendicular, 90 degree angle from any point to a line. By using these three simple techniques, we can generate those four and more because you can combine any of these. Each time you start, it will create a different pattern, a unique pattern because of where the points are located. In order to be able to make any of those and have fun decorating them as well. This is what we're going to need, some watercolor paper, any shape or size, tracing paper, some cling film or plastic or bubble wrap, You're going to need paints, masking fluid, possibly some salt or anything that you think can create texture. Of course, we can't construct anything without having a compass, a pencil, a ruler, and then we need a waterproof and a metallic paint as well. Let's get started. 3. Halving a Line Segment at a Right Angle: Before starting to learn how to do Voronoi diagrams and create our polygon teslations, we first need to learn a very simple geometric concept that is used over and over in geometric art. That's how to take any line, random line, and be able to split it in half at a right angle. This is called a perpendicular bisector, perpendicular because we're going to find the right angle, bisector, because we're going to split the line into two equal sections. The line segment has any length. All we have to do now is open our compass to a length that's larger than halfway. And start from one end of the line. Instead of drawing a full circle, we're just going to draw an arc from below to above somewhere along the middle. Without changing the radius, we just do exactly the same from the opposite end of our line segment and make sure they cross. If these aren't long enough, we can go back to extend them. The two points that we've just created by intersecting these two circle arcs shows us where to draw the line. Which will be at a right angle, so this here now is a right angle, and these two line segments are equal to each other. The reason why this works is because by drawing a circle to there, this distance and this distance, and this dice and this distance, were equal, we've created essentially a rhombus, and the diagonals of a rhombus are always at a right angle to each other and always split each other in half. Now using this simple concept, we can start drawing noi diagrams. 4. Drawing the Voronoi Diagram : A Voronoi diagram, essentially, all it does is if we have two different points in space or sits, let's say one here and one here. What the Voronoi diagram does, it partitions the surface into two areas that shows us which side we are actually closest to. We're going to start with two, two arbitrary points. Now we want to split the space into two parts. Because for example, if I pick this point here, do I know whether I'm closer to this point or this point? This is how we're going into the side? We're going to find the perpendicular bisector of this line segment. However, I'm not going to draw the line because the diagram will get too overloaded. Just like before, open distance as long as it's small and half, but make sure that it will cross within the boundaries of your paper and draw an arc and do the same from the other point without changing the radius just as before, there you go. Now we know which two points the line will go through. This will be the perpendicular bisector to the line segments that we haven't actually drawn, the one through those two points we had. The points are going to determine the shapes in. If we only had two sites, then we're only going to have two polygons. Any point in this region is closer to this point than this point. An point in this region will be closer to this point than that point. Any point on the line is equally distanced from both. And that's how it diagram is created. Now, of course, we're not just going to split the page in two, we want to add more. I always recommend just add one at a time. I'm having to go around watercolor piece of paper, see how it goes. Of course, we can rotate it however we like. I'm going to start with two sites. The first thing that we need to do is find the perpendicular bsector of this line segment, if the two points we drew were the end of the segment. We're not drawing the actual line, but just finding its bisector. Two arcs from both ends through the two intersections. And we've partitioned our Voronoi diagram into two regions, each of which is closer to the site. Next, we're going to add another point. We're going to find the perpendicular bisector of the new point with the existing points. Let's do the nearest one first. We can go back to extend. The next line goes through these two points, and we can stop at the point of intersection, and this line now becomes unneeded. Now we need to find out where the perpendicular bisector between these two points is because something is going to be added down the bottom as we're looking for three regions. Open wider to make sure that we th at least half. This looks good. Little short. It often happens and align through the two new intersections, but only draw from the vertex down. Generally, when we build our Voronoi diagram, all the new vertices we find, which are the boundaries of the regions. There should always be three line segments coming out in different directions. Let's add a point here. Same as before. Now, when you are close to the edge of the paper, you need to be careful that you have an arc long enough to cross. It must be more than halfway but short enough to cross, not to go past that edge. Let's take this and stop at the new intersection. Here we have two lines. This is unlikely to be needed anymore this part. So there should be another line going across. Now I suggest that we take the new point and find the perpendicular bisector with this point. It looks like this point is closer than this one. Again, a bit of a wide radius. Carefully locate where the intersections were. The top one is easier to see bottom one and only draw the segment from that intersection. At this intersection, there are now three lines, but now here the ramp. As you can see, this is definitely not the middle, can delete this part. If I find the perpendicular bisector of these two, I can predict it's going to start most likely from that point there because we're looking for a third line coming out of that vertex. New intersection there and here. Then it's coming down through here. Yes, that makes sense. Now we have four regions and four sites in them. I'm going to add one here. It's a little bit bare. Let's add one here, and we will need to find a few others. I always start with the nearest. It's obvious that we're going to need that. Some of the others we might not. And let's have one more here. Hal way between these two. One here. We are going to fix this here by looking at halfway between these two. And this one here between these two is going to be through here because these needs three each. And that can be thud. Let's just add one more down here. And that makes sense that it goes through here. You can add more, you can stop at any time, but I'm quite happy with the size and shapes of these. It turned out quite regular. A we're going to keep these lines, and then the rest will get erased and that will be the design. Before we move on to decorate it, however, we can trace three more designs from the same, so get your tracing paper rey. Go. 5. Drawing the Triangulation Design: This is the Voronoi diagram completed, and I've put a piece of tracing paper on top. Luckily, we can still see the sites we started with. These were the original points, and around each site, there's a region where the edges of each region meet, there's a vertex. All the vertices we found in the Voronoi diagram have three edges meeting there. Now what's really nice, a secondary design, we can do directly from this grid. It's actually what we call the dual graph of the Voronoi diagram, and that's simply a triangulation, because all we do now is connect the sites that we started with. Since every point is close to two others, we're going to create a chain of triangles. The Voronoi diagram can have any shapes or polygons inside, where the triangulation is strictly made of triangles. We connect all the sites, but it makes just triangles. You'll see how easily that works. Now, if I draw this line, that was the original two points I believe we started here. This is the distance between the sites. The diagram we're drawing now focuses on the distance between the sites, whereas the Voronoi diagram focuses on the area which is closer to the site. We just connect all the sites together to create triangles. Looks a bit like a sea shell, doesn't it? Now we don't just want the triangles in the middle and we don't have enough points further out. That's fine. A bit of creativity here. What I like to do is to take each of the vertices on the outside. These are the sites actually and just connect them each with those at the edge of our paper. Just like this. That means that we have triangular shapes all the way to the edge of the paper. I connected this to there and there. Some lines might look like they lie on a straight lines, some might bend slightly, might look a little bit like a star on the end. That's okay. Generally, it's a design of a variety of different triangles. All along the edges, there's also triangles, and we have all these vertices. When we trace this back onto another piece of paper, we can see where the circle will match. Now, you could have that as a standalone design in its own right, or you can leave it as overlapped over the Voronoi diagram. Not only can you leave it overlap, you could rotate it and create a new pattern. It doesn't have to be in that order. If you rotate it, it will create other shapes. It depends on how many shapes you want inside your design. Do you want fewer bigger shapes, or you can combine the Voronoi and the triangles together. It will make this. This is one example. Don't take the tracing paper off just yet because we're going to use both of these layers to create the next one, which is going to be circular. 6. Drawing a Circle around a Triangle: The third variation is going to be based on circles, and the first type of circles we're going to learn, are called circum circles. That's short for circumscribed circle. Start with any three points on your surface. We want to find a circle which surrounds all these three points. It goes through exactly those three points and it's always possible with any three points. Essentially, this is a triangle. Whether we draw the lines or not. We will just for clarity in this case. How do we find where the center of this circle is going to be? We can draw it. I will be somewhere here, we can't guess it. The way to find it is by finding the perpendicular bisectors of the lines of that triangle. Let's start with this line on the bottom, open your compass a bit more than halfway, T arcs as before, exactly the same concept as we have been using and draw a line through the two intersections and extend through the other side. Of the triangle. Just make sure it's long enough. Next, let's do this side. I can use the same. Radius. These are the two intersections. Extend to make sure these cross. You won't be surprised now that I told you earlier, when you draw the third bisector perpendicular bisector, it should go through exactly the same point. In which case, it means we don't need all three any two or enough, but just to show you that this should work through here and here. Extend all three cross at this point. This is going to be the center of the circle which goes through all three points. What is the radius going to be then? Well, the distance between that center and the three vertices be equal. Just measure, make sure it's going to go through all three, and there you have a full circle that goes through any three points. That can be really useful in any geometric design. If you have three points and you know you want to make a curve that connects three, even if it's not a full circle, this is how you do it. Connect the three points into a triangle, find two of the perpendicular bisectors and find the center by crossing. 7. Drawing the Circumcircles Design: As you saw on the previous video, we can circumscribe a circle around any triangle as long as we can find out the perpendicular bisectors of the edges of that triangle. In fact, two edges are enough. But what's the good news here? We have already drawn the triangles. And we know for every triangle that we've drawn, we have already bisected it by drawing the Voronoi diagram underneath. Let's take this triangle, for example. This is a triangle. That edge has already been bisected at the right angle by this line and the same for this edge by this line and this line. That means that we already have the center of where our circle is going to be. We don't have to do any extra working out because the work has already been done. This is the center the radius should be to the three points of that triangle. Let's draw it. There we have it. There's a circle. We didn't have to do any more working out than what we've already drawn. Now, see this triangle here, it's interesting because of the shape, its center is very near the edge. In fact, sometimes it is possible that the center of the triangle could be outside of the triangle. It will just depend on the shape. You look at where do the three perpendicular bisectors of the edges meet here and then extend your radius to go through. The three vertices that make up that triangle. This one is not as accurate. I think I just need to be a bit further down. Here we have different overlapping circle. That's what I like about this design, different overlapping circles, but not random. They have a structure underneath. Let's move to this one. There is the center. This is even bigger by the look of things. Let's just check that it crosses points. Yeah, I'm happy with that. Great. There's the next one. Here is the center. This one, sometimes they have the same size or this one slightly bigger. I think the center slightly lower than what I went for, but that's okay. There's one more here. It's okay if some are partial and some come outside because ideally we want something on the outside. There's one more here. You could of course just add a few extra arcs from here, just arbitrarily. For example, we could go from here and draw an arc through this point, for example. We could do the same from here to this point. H. As I feel that not that many other places need more arcs, perhaps one more here and then I'm happy to stop. Yeah, I'll stop. Now, as I said, you could just find one more perpendicular bisector, let's say this line, just by drawing the two arcs, and see where it crosses and do this one entirely inwards. But it's nice to have some unfinished ones on the edge. Now this is the circum circle design all complete. As you can see, it can work on its own as a design in its own right and painted really interestingly, or of course, you can keep it as part of the straight line designs. For example, you can combine it with just the triangulation or just the Voronoi diagram. That might be interesting as well. 8. Drawing a Circle inside a Triangle: Final variation is also made of circles, but not the circles that surround any of the triangles that we have, it's the circles that fit inside of the triangle. In order to find the in circle, we need to learn two more very simple concepts. We know how to bisect a line. That's the first thing I told you. But now we need to learn how to bisect an angle. What does that mean? Well, an angle simply measures the amount of turn from one direction to the next, how much is this turned? The angle bisector, what it's going to do is going to find the line that will split it in exactly half. We don't need to measure it, we're very simply going to construct it using our compass. Similar to before, we're going to just draw a few arcs and see where they overlap. Starting from the vertex of the angle, where the two lines cross, Just draw an arc that will cut through both lines. Then that means that these two distances are equal. From the two intersections we created, just cross in the middle where two more arcs are going to cross. And that shows us where the third line will go from the vertex of the angle through the new intersection we created in the middle, and there we have it. These two parts now are equal. How do we find the in circle? We need to find the angle bisectors. Just as before, we only need two out of three because they are going to cross at the same place. I'm going to do start from the bottom and split this angle. Go to make that a little shorter. Cut both lines with the same arc, and then from the two new intersections. Just two short arcs that will cross in the middle if they're not long enough, extend them, and here we have the bisector of this angle going through the intersection and through the vertex. Extend through the other side of the triangle just to be sure, and now we're going to do exactly the same on this side. You can change the radius on here, doesn't have to be the same. Go to make it slightly longer since it's a bit of a smaller. Opening, and then from these two intersections, cross through the middle and extend this line through the vertex and the new intersection. From here, Yeah. Where do these two lines cross? It's exactly here. If we did the same thing through this angle, it will also go through there. You could do that to try and verify this or use it just as a way of checking your accuracy if you like. But for now, I don't need to. Now we know where the center of our inscribed circle is. Unfortunately, we don't know yet how far to open the compass. We don't actually know the radius. The radius doesn't go through those points. Those points here are irrelevant. We just needed this. We could try and guess it like this by just hovering around and seeing where it's going to touch. But we don't need to guess. All we have to do now is find where any of the radii of this circle will touch any one of the lines. The simple rule here is that the radius of a circle will make a right angle with the point of tangency to that line, which means we just need to find the right angle, in other words, the perpendicular line from that point to any one of the three lines, and I'm just going to use this one because it's wide. The last fundamental skill you need to learn is how do we find the perpendicular of any line from any point, not necessarily the one that splits it in half, but any other point. This is what we do. This is called the perpendicular from a point, not the perpendicular bisector, because quite clearly we're not going to split the line in half. From the point we have, we need to open the compass wide enough that it will cross the existing line in two places, one here, one here, or just one continuous arc. Now we have two new intersections. From the new intersections. Just check the distance here is the same. Create an arc on the bottom, from the opposite intersection, another arc on the bottom, and you guess that through the point and the new intersection. This line here creates a right angle to the horizontal, but it doesn't split it in half. Why is that useful? Because this is how we're going to find at what point our circle is going to be touching this line. We know the center of the circle. That's the point, and we know the line we want to find the perpendicular two from here. We're going to start from the circle center or that point above the line. Open the arc, so you know it's going to cross your line twice with the same radius from the two new intersections, draw another two arcs below. And now align the circle center with the new intersection that we found, and just draw this vertical, which is at the right angle. This is the in center, that's the center of the in circle. This is the point of tangency. In other words, we've just found the radius for our inscribed circle, so now we can draw it. Start from the center and align the radius with this point here. Now, you could do this for all three sides, but you only need to do one of them. A little bit wide. There we've inscribed the circle. 9. Drawing the Incircles Design: The final design is slightly trickier because we need to use two more concepts just as in the previous video. For this one, we need angle by sectors and then a perpendicular from a point. I've removed everything. We just need the triangles because the other two designs aren't helping, and I'm just going to put another piece of tracing paper on top. Let's start with this triangle here on the right, it's fairly big. We need to find two of the angle by sectors. A two. I'm going to go for this one to start with. If you remember from the previous video, from the corner and arc to cut both segments. Then from those new intersections, another two arcs in the middle until they cross. We know where to draw the line through. I ruler, align the vertex that we used and the new intersection and extend more than half. Now we know the in circle is going to be fully inside the triangle, therefore, we don't need to extend beyond any of the triangles. I'm going to do exactly the same from this corner inwards into the same triangle. Cross the two arcs, cross the two edges, two more arcs, a straight line through here and here. It's not an easy technique to get super accurate by hand, but it is definitely very valuable geometric technique. Now we know that the center of the circle is going to be here. Now we need to find the perpendicular from this point to let's say this line so that we know exactly where the circle will touch without having to guess. Again, with the compass, from that center where the 22 bisectors have crossed, extend to make sure that your arc is going to cross the edge of the tri and go in two distinct places. And from those places and arc below from and another one to cross. Now the reason why these are tricky. This is a very shallow intersection and it might be difficult to judge exactly the millimeter of where they meet. Don't be discouraged. We're still going to do exactly the same what we need it to do. We're going to align through here and the center, but we're not going to go beyond. In fact, we only need that point there, but basically what we've just drawn now is the radius up to here from there to there, is the radius of the circle that we're looking for the in circle. As I said before, you could guess where your circle touches. Once you know where the center, that is really the most important part. This should just touch through our triangle. Let's try it. And there we have it. All we have to do now is do that for all the triangles there. I added a few on the outside as well and deleted some of the arcs so that it's easier to transfer. It's just those bubbles. It's a design on its own. However, they're all separated, unlike all of the other designs we've made. I recommend that you overlap it with the other circles in any orientation or the oi diagram or see how it goes once it's been transferred. 10. Painting the Voronoi Diagram: Okay. We're ready to decorate our first design. That's the Voronoi diagram we created earlier. I'm going to use some very bright colors and decorate each region in a similar shade. I'm also going to use some salt to give each color a bit of texture, and I'm going to first wet the paper and then add the individual colors. For example, I'm going to start with the middle. I'm going to put some water on. I like using a flat brush for this. It's ok if the colors bleeding to them, but I'm going to do them separately. I like to apply the color with a big brush. That way. My color isn't going to dry too quickly. I'm going to start with violet, which is my favorite and put that in the middle. I'm going to add a bit of water here. I really like how the water distributes the pigment. Just trying to make sure that the edges are saturated even though we are going to outline the corners at the end. What I'm going to do now is put just sprinkle a few sea salt grains. As they are still on there, just going to drop another couple of bits of color, and they're going to go the gaps that the salt is creating. I'm going to do something similar with the next color. Next, I'm going to go here and I'm going to use some hyacinth. My magenta shade. This one. As I said, I don't mind that the colors are bleeding a little bit into them. In fact, I like that effect on the edges. I'm ready for some salt. Just add a few bits here and there. I feel like I might add a little bit of this color here. And a little bit of this color here. Next, I'm going to go for some turquoise, for a different vibe. I actually do really like the combination of purple and turquoise. It's one of my favorite color combinations. Some salt. A couple of purple drops. Next, I'm going to go for cis, which is cherry red. Looks similar to that, but it isn't. That's great here. Let's add a few turquoise loves and salt. Notice here I added the turquoise before the salt, so it will create a slightly different pattern, I think. Next, I'm going to go for, which is the red vibrant red. I'm going to add a bit of red on here as well to break up that turquoise. Okay, salt. P drops of turquoise. And p drops of purple. Next, I'm going to go for blue. Lt Marie. These two are looking a bit too similar now because of the addition of the other colors. Probably put a bit too much in clusters. But that's fine. It's unpredictable and fun. And that's what we like about this. Might be unusual, but I am going to drop a bit of orange into here. A little bit of red. A bit more here. I want it to be distinguished from this one. Then the final one, I'm going to do an orange. I'm actually going to use this brush for this final color. So some more cherry red. I touch more purple in the middle because I feel like it's not quite as purple as it started off with, and I wanted this to be more purple and to have different color to one. Then I'm just going to drop a couple of turquoise bits into the orange. So interesting. I love the geometric structure, but with these abstract coloring techniques. Now, anywhere on the edge is where I feel like it needs to join in, I'm just going to drop a bit of water. That's too much. I'm going to use a bit of tissue there. Gently, just join in. As I said, we're going to outline this with a golden pen, but we don't want to risk having any gaps. Quickly go with some water along the edges, just to tidy things up. Just drop water, don't even worry about merging the color. I feel this means a little bit of dabbing off. Very vibrant and yet lots of textures. Allow the salt to completely do its job and for this to completely dry as it is before we can scrape off the salt and then outline the edges with gold. The Voronoid diagram is now all dry. I do feel like it needs a contrast. I'm going to go with a thicker pen and it's copper. Hopefully that stands out. You can use anything metallic, silver would look great, gold, but I feel like using copper. I'm going to use my ruler. I flipped it upside down. The metallic doesn't smear too much on the ruler, although some will, so we'll have to just wipe it off afterwards. Just go over the boundaries for the nice and vibrant. I think I like the thicker one better for this design. So I'm going to repeat some of the thinner lines, make them look uniform. You should be able to clearly see the lines from the waterproof line underneath. Yeah, that's nice. Wow, there we have it fully complete, worked out, painted, outlined, beautiful Voronoi diagram. 11. Painting the Triangulation Design: For the next pattern, we are now going to transfer the triangulation design that we traced out earlier. Because I'm using a circular paper. It doesn't actually matter which way around. If you're using a square paper, you can decide which way around you want it. I'm going to just secure the tracing paper. Okay. I've turned over the tracing paper so that the graphite is going to make contact with the paper. This is the orientation I've chosen. I'm simply going to go over the existing lines with a ruler and a pencil. Okay. Here we go. Here is the trace design. I do feel like I want to add three more segments just here because these are quite nice and flat, but these are fairly big. You don't have to. Just to feel like it's more of a complete design. We're going to paint the triangulation design now. I've decided to go with lots of different greens and I'm going to just scatter a lot of greens around the whole thing. I'm going to wet the page again with my flat brush to make sure that each part of it is saturated. Now I'm not going to worry about where the boundaries of the triangles are. I'm just going to randomly put the greens. Later on, we're going to outline the boundaries to give it that extra structure. I'm going to start with some brilliant green. And randomly scatter it around the page. Paying particular attention to the corners. I like the, the boundaries of the page to be saturated. Next, I'm going to go with some peak green, which is a lighter shade. Be careful with the edges. There's always going to be more water that gathers there. That's normal because of how the paper is going to move. Next, I'm going to add a bit of a darker shade of grass green. Create shapes if you like, or just allow the water to distribute the paint in whichever direction. Want. Create a bit of a shading here. I'm going to add a bit of turquoise now. It will give it a bit of a blue blueness to it. But of course, the majority will be green, so that will be. Going to go over some peak green again. That was the original the light shade. It's like breathing through in a forest. That's what this makes me feel like. I'm going to add a bit of brilliant green again. And last but not least, I'm going to scatter a little bit of neon green. That's just to give that extra little brightness of the whole thing. Going to try and spray some of it. I do like this effect as well. That's why I wanted something brighter, so it's a bit more contrasting. You could do the same with a bit of white as well. But I think that's what I was after. This effect. It's interesting. And abstract. I'm completely going to leave that to dry. Now if you think that's too intense as a color, you could just cover it with a bit of tissue and just dab some of the color off. The green is all dry. It's beautiful. I really like the contrast, as you can see, I took away some of its vibrancy, but it's still quite dark, but we can still see the lines. Now for the frame, I'm going to go for firstly a thick black marker. But then I'd like a bit of shine. I'm going to put some silver around the black so it looks like a frame above the background. I'm just going to go over the lines with a thick black marker. Now, I'm going to outline on either side of each black line, meaning basically, I'm going to align the inner side of the triangles in silver. That's the triangulation design complete. I like it. Wonderful. Next, we can turn to the round designs. 12. Painting the Circumcircles Design: We're now going to transfer the circum circle design. These are the larger circles that we're going ad the triangles earlier. Luckily, the points where we've used earlier to draw each circle, we could still see should be able to see and even feel through your tracing paper when we traced it earlier. All I'm going to do now is find carefully the center of each individual circle and make sure that the graphite is pointing downwards and just draw on top of each circle. Hard enough to make sure it transfers underneath, so we can paint that. What I have now, I haven't outlined this in a waterproof pen unlike all the others. What I've got here is some masking fluid, and I have this silicon taper end which I can use to actually draw with mine luckily fits in my compass. What I'm going to do is outline all the circles using the masking fluid. We just need to let this dry out before we can paint. The masking fluid is now dry. I'm just going to paint a few random colors as I go along in each circle. I'm just trying to anticipate which color will overlap with which. I'm not going to make this very wet, just trying to make it uniform, but it will be very nice once we've rubbed off the masking fluid later because it will have hopefully that definition that we're looking for in between the circles. I'm going to wait between layers for each of them to dry before I start the new colors. Next I'm going to do some orange. I'm going to do some orange here as well. Hopefully, it should dry well. So orange here as well. So I'm trying to overlap with a different color, but not with the same color. That's kind of what I was envisaging. Okay wait for the orange to dry or use a hair dryer. Next, I'm going to have some red starting here and here. Have to go as fast as you can really, which I'm not necessarily doing well, but it's so that the colors don't get transferred, but stay more overlapped. I will stop with this color now. Next I'm going to add the slightly darker cherry red. Bit shed here and there, but be fine. I'm going to start with this part. Oh, this color is just glorious. I'll make that smaller circle. It all laps pretty much everything. And then this one here. Now I'm going to add some hyacinth on the outside. All these colors look nice together. This isn't the best brush for this. It's pretty, isn't it? I'm just considering now whether I should paint one of these here. It feels like a lot. Whether this one or this one. Which means that yellow will disappear. But it will still create different shades. I'm going to do this middle one in this color. Okay, the colors are now dry, and I'm definitely going to put this under a heavy book after in order for it to flatten a bit more. But for now, that's great. But let's have a go. So I have this special rubber for masking fluid. This actually worked really beautifully just as I wanted it. Some of the paints spilled over to the next one, but that was more me with the brush rather than the masking fluid did its job really well. There's only one obvious point there. The reason why this happened is because the point of the compass for one of the circles had to go on there several times after I had put this, I remember having to put the pin on the masking fluid, and of course, that's made a hole in the masking fluid. Some of the paint has gone through. That's not a problem. Just a little bit of white will fix it, which will probably disguise the actual point as well. The little hole. Even though they never bother me they're part of my normal geometric art, a little imperfection there, a little bit here where either the pencil hasn't covered or some of the brush transferred a different color. A couple of tiny finishing touches. But I'm very happy with that. I really like it where at some points different overlaps happen. There's a tiny bit of yellow there on its own, which is great. There's a bit here. But I always say, this can work a little bit, but nothing can really in my view, nothing can fake the whiteness of paper. That's why I don't want to get carried away doing this in white because the whole point of this for me was I wanted the white of the paper. Yeah. We're all done. 13. Painting the Incircles Design: We are now ready to outline our last design, the bubbles, the in circles, they're the only design that isn't actually interconnected. We're only going to trace the circles and ignore all the construction marks. I'm going to start with the biggest one and follow them in this round order. Then I'm going to do the side ones. Make sure you press as hard as you can, in order to transfer below. And remember we added a few around the edges. I highly recommend when you finish playing with this to try overlapping the designs. I particularly like the two circular designs together. All the small in circles always fit inside all the larger circum circles if you align them in the way they were constructed, or you can overlap them totally randomly at a different angle and the Voronoi diagram, if you want to combine straight edges, and circles, that's another good idea that you can try. I think that the in circles look better with the Voronoi diagram rather than the triangulation, simply because with the triangulation, all the circles are exactly fitting in each triangle. It's a bit too regular. Was it's not as obvious and it has a bit more of a movement and dynamic if it goes together with the Voronoi diagram. Was the triangulation goes really well with the large circumscribed circles. If you feel like you need to repeat some of them, that's fine. Yeah, I'm happy with that. This is what it looks like if you combine it with the larger circles. At the moment, they are crossing each other, meaning they're not in the same orientation as when we first constructed them. If you see the three big circles here, they would be within the three bigger circles here and the three closest ones. This is likely to have been the original and you have a circle inside each of the larger circle. But in any case, I think they look fantastic overlap together. Now, because this really reminds me of bubbles, I am actually going to use some bubble wrap to try and create some texture and I see this as because they're separate shapes, I feel like I want to paint the whole thing and then later emphasize the circles. I'm going to start with wetting the paper. Then I'm going to go with lots of different blues because it just reminds me of water bubbles. Then the first layer of the background. We're going to use the bubble wrap to create some texture. You can use cling feel more or anything else that will create texture. However, I think because of the circles on the bubble wrap, it's just that extra thing added. I'm going to start with my medium blue color. And I really don't mind in which order I place the color. The water is curling the paper. That's completely normal. I'm going to go with deep blue now. I'm going to emphasize the edges in particular. Finish. Doing the edges because I want to emphasize the roundness of the paper as well. So even if it's not very dark shade, I don't want any white spaces around the ends, if that's possible? A few more shades here and there, just to mix it a bit. It has a few more layers in depth. We do want this to be quite strongly colored because the plastic is going to take some of that off and the intensity will not be as strong. We won't see the texture very well if we don't saturate the paper with color. Finally, a little bit of bright neon blue. In a random completely random order. Now I'm just randomly going to place this, push that down. It's okay if it's torn apart. In fact, it will probably create an interesting shape. If if I torn that apart and it wasn't all joined together, but it was torn and it had gaps in between, see the effect it already creates. Let's leave it to dry and see what happens. I'm going to reline them with silver because they're very difficult to see. What the silver is going to do and help us with Lao as well is that when we paint inside the bubbles, the paint marker, the silver paint should help the paint not come out of the boundaries. Of course, they can be relined after we paint the bubbles, and layer them on top as well. Now, I'm going to paint with metallic silvery blue or metallic blue. That should cover up the previous layer. Of course, we don't have to cover the entire bubble. Let's start with this one, for example. We can just do the outline, see how you feel. Very nice metallic accent here. You could of course add silver or gold if you have that paint. I feel like pushing the pigment towards one end of the circle to give it that almost bubble look a bit like here, it's sedimented. Of course, that will be a bit tricky when the paper isn't completely flat, but it should be okay. Should draw in a very interesting way. And the in circle design number four is also complete. 14. Conclusion: I really hope you enjoyed learning about these four variations that can be all created from the same grid. The main thing, of course, is the foi diagram, and then anything related with it, the triangulation, and the two circular designs that can be inscribed or circumscribed around each of them. I hope that you get to experiment by overlapping different designs with each other and enjoy the idea that you can create as many different ones as you like each time. I hope you enjoyed exploring some of the fun decorating ideas that you've had, and you can take this forward into any of your own art. Please don't forget to share what you create in the project section, and tag me in an Instagram so I can see your artwork.