Good morning I’d like to invite you to a film different from the previous ones. . There won’t be any spectacular ideas or fireworks here today, but you can expect a collection of rules that’ll come very much in handy with flat or perspective drawings. The drawings you see here are examples of what I’m about to show you in raw form, put in practice. We’re going to divide this film into pieces right away. You can see the time reference on the screen. Now, I invite you to join me. Let’s start with drawing a square. I can measure off an equal length of all its sides with a pencil. Today’s drawing is sponsored by the letter D, like the diagonals. I can find the centre of any plane with their help. Drawing any diagonals will come in handy for us today, more than usual. We can project the centre to the square’s sides to get their middle points. We can take the distance from the square’s centre to the middles of its sides and move it to the diagonals. Let’s check if those points connect vertically and diagonally. If yes, we can be sure we’ve got 8 points equally distant from the square’s centre. It’s like they’re just asking to be connected into a circle. Here comes the 1st drawing tip – – our hand makes smoother arcs in some directions than in others. For a right handed person it’s easiest to draw the curve in the upper left quarter. The arcs in neighbouring quarters are less convenient to draw, but the last quarter, in the lower right corner, is the worst. It’s easiest to just turn the paper and make use of our hand’s natural gestures. Unfortunately I don’t have this luxury while drawing with the camera. Remember these actions we just made. We’re going to be using them intensively now. Let’s start with the basics that many of you may already know from this channel. I promise it’s going to get less typical further on. Let’s draw a small square with diagonals. Let’s say we want to stick another identical one to its bottom. To do that we need the middle of the square’s side and we get that by using diagonals. Let’s extend the square’s sides in the direction that I want to copy it. If now I draw lines coming out of the square’s corners and going through this square side’s middle, the points where they intersect with the square’s extended sides will create the corners of the new copied square. We can also copy the rectangle composed of these 2 squares. We already have the side’s middle so we only need to extend these sides and lead the corner lines to them. And voila! We get the rectangle’s copy. We also get a square with the surface 4 times bigger than the initial one. What if we want to copy a rectangle rectangle so that it starts exactly where we plan and it isn’t attached to the original one? Here again, the sides’ extensions will come in handy. Between them and the new rectangle’s 1st side, a rectangular area is created. We can find its centre with diagonals. That’ll also be the centre of the whole copied arrangement. Now it’s enough I lead lines through this centre, starting from each of the rectangle’s corners. A new copied rectangle will be made from these lines intersecting with the rectangle’s extended sides. From dividing into 3 and higher we can also try our hand in crating analogical regular polygons. Let’s start with 2 squares. The diagonals show us the centre, of course, and they help us divide the squares into halves. Let’s add the diagonals of the created halves and find the points where they intersect with the diagonals of the whole square. These points moved to the sides will divide them into 3 equal parts. For drawing an equilateral triangle in a square we’ll mostly need two big arcs with the radius equal to the square side’s length. I can move this radius with a pencil to any straight lines lines that come out of the corner, which is also to arc’s centre. For the whole to look orderly I copy the radius to square’s diagonals and its halves’ diagonals too. But that’s a purely aesthetic matter. A similar arc must be analogically constructed from the other point. Then the point of arcs’ intersection gives us the vertex of the equilateral triangle, similarly to the arcs’ lower beginnings. Even though we had a new rule for dividing into 3, we don’t need anything new when dividing into 2 and its exponentiations. It’s repeatable. We find the centre of the square with diagonals and divide it into halves. We find the halves’ centres, also with diagonals and thus create a division into 4. It might not be very extraordinary but it’s very useful for architectonical drawings. Finding smaller squares in a bigger one is similarly useful. The most obvious thing is to connect the middle points of the big square’s sides to get a smaller square inside of it. Its sides are already divided into halves by the big square’s diagonals so we can find an even smaller square. Its sides are also already divided in half by the segments connecting the centres of the big square. And so on. Some years back, an exercise based on this rule, was on the entrance exam to Warsaw Faculty of Architecture, where I studied. In architectural drawing it’s equally important to be able to construct a square rotated around a circle or inside of it. We’ve just drawn a square created by connecting the middle points of the big square’s sides. But when we have its circumcircle we can draw other identical squares rotated around a common centre. I’ve used the points of intersection of the circumcircle with the big square’s diagonals as corners. We’ll get back to that when talking about the octagon. Division into 5 requires some more work. he Thales’ theorem or the intercept theorem will come in handy now. This means we’ll 1st create a working segment composed of 5 even parts and then move them to the segment we actually want to divide. The square’s vertical side is already divided into 2 by diagonals. We can also get the division into 4 by using smaller diagonals. Next, using copying, I move the ¼ long segment downward and thus get a working segment composed of 5 even parts. I now have to draw the diagonal of the rectangle created from the new segment and the square’s side. Horizontally, I move our division into 5 onto that diagonal and then I copy it vertically onto the side of the square I want to divide. These kinds of divisions are helpful when drawing antic and classical architecture. There is some fun with the pentagon too. In the upper half of a square with diagonals and an incircle – I find the vertical segment’s division into 3, and in its lower half – into 5. This is a simplified method, which is not mathematically perfect but assuming we’re drawing by hand, we don’t have to worry about such small inaccuracies. When we have the right divisions, we move point 1/3 and point 1/5 horizontally onto the circle. The connection of these points with the upper point of contact of the square and the circle will form a pentagon. There’s another, different method for constructing a pentagon. We’re going to need the middle of the inradius. We connect that point with a neighbouring point of contact of the circle and the square. Next, we draw the median of the vertex created by the 1st chosen radius and the line going to the point of contact. This median intersects with the inradius, giving us a point which, projected to the circle, gives us the 2nd vertex of the pentagon. You have to repeat this whole action all around. This method is rather unpleasant. I personally use the previous one way more. Dividing into 6 is easy-peasy compared to the previous one. We actually have to put together 2 divisions with diagonals. 1st into 2 and then, from created halves, into 3. The only thing I might add here is that maybe it would be easier to create the division into 3 in a square’s quarter and not in its half. For comparison: on the left side I divide everything from top to bottom and on the right side I only divide the upper quarter. The final outcome is the same same but in the 2nd method the lines are shorter and they intersect with better angles, which gives us a little bit more of drawing precision. Drawing a hexagon is also easier than drawing a pentagon. Let’s divide our square into 4 even stripes. We’re only interested here with the lines distant by ¼ from the top and bottom side of the square. The points of contact of these lines with the circle and both top and bottom contact point with the circle mark a hexagon’s vertex for us. We can also use this method to draw an equilateral triangle by only connecting together every second vertex. We’re going to skip division into 7 for now. I’ll say why at the end of this film. As I’ve mentioned before, dividing into next exponentiations of 2 is nothing else than finding halves of next halves. We can make it easier for ourselves by moving the created divisions horizontally onto the proper diagonals. We’ll draw an octagon by using a circle and squares from the drawing where I’ve talked about division into 4. It’s important that the 2 squares are placed in a 45° angle to each other. That means the vertices of one have to be placed on the circle and the diagonals of the big square and the vertices of the other have to be on the circle and the middles of edges of the big square. We can connect every next point on the circle and that’s how we get an octagon that is tangent with its vertices to the initial square. We can also strengthen the joint part of both squares and get an octagon, which sides are parallel to the square’s sides. We can also create its bigger version by drawing lines parallel to the square’s diagonals that are tangent to the circle, where the square’s diagonals go through. This way we get a big square rotated along the circle by 45°. The part where it’s joint with the initial square is also an octagon. This method comes very much in handy when drawing i.e. medieval architecture. We can also use the existing divisions to make more divisions, just as we did with squares. That’s it when it comes to the divisions useful for architecture drawing. I also recorded dividing into 7, 9 and 10 but they are not as useful and rather rarely used. I don’t want to scare anyone off with them so if I show them, it’ll to be in an additional film. Let me know in the comments below if you’re interested in watching that. And in the next film of this series I’ll show you a more practical use of the methods presented here, in perspective drawing. I encourage you to subscribe to our channel and see you soon!