Working out the order in which a figure has to be drawn

0:00Last time we copied five drawings freehand, and they came out roughly right. Then we did it again with a ruler and a compass, and something changed. But notice what the instruments actually gave you. Not a steadier hand. They gave you the ability to make a length exact, and then repeat it. Here is the thing nobody warns you about, though. Once the shakiness is gone, a completely different difficulty is left standing, and it is not about drawing at all.

0:34Take this figure. A person, roughly. Before you draw anything, break it into pieces. There is a body: a four-sided outline whose top edge has been replaced by an arc curving inwards. And there is a head: a circle, with a short straight segment below it. Two components. That is already useful, because now you are not drawing a person. You are drawing three things, one at a time, and the only question left is which one goes first.

1:09So which one does go first? The instinct is to start at the top and work down, or start on the left and work across. That is not the rule. The rule is: start with whatever needs nothing else. Look at the body outline. Its four corners are wherever you decide to put them. It needs nothing. Now look at the arc. It runs between the two top corners of that outline.

1:36So the arc cannot exist until the outline does. The corners have to be on the paper first. And the head sits on top of the outline, so it needs the outline too. One part needs nothing, and two parts need that one. Which means the body is not just a good place to start. It is the only place. Let me draw that as a picture, because it is the picture this whole video is about.

2:06A box for each part, and an arrow meaning cannot be drawn until. Body, arrow to arc. Body, arrow to head. Now count the orders that actually work. There are six ways to arrange three things. Only two of them respect those arrows. Body, arc, head. Or body, head, arc. The other four are impossible before you pick up a pencil. So the arrows are not a suggestion about tidiness. They rule out two thirds of everything you might have tried.

2:41Now the step that is genuinely hard, and it is the middle one. You have the outline. You want the arc across its top. What do you actually need? Two things, and neither is optional. Somewhere to stand the compass tip, and how far to open it. And here is the awkward part. The two endpoints do not tell you. Watch. Same two corners, and an arc through them. Now a different opening, still through the same two corners. And another.

3:13Every single one of those is a genuine arc joining exactly those two points. The endpoints fix where the arc starts and ends. They do not fix how much it bulges. So this position is not calculated. It is judged. You try a radius, look at it, and move the tip until it sits right. That deserves saying clearly, because it goes against what school usually implies. Estimating here is not sloppiness, and it is not a shortcut. It is the intended method.

3:49The figure genuinely does not determine that arc, so there is nothing to calculate. Compare it with a case where the figure does determine everything. Because that difference — determined or judged — is a thing you can work out in advance, on a sketch, before you commit any ink. Here is a figure that makes the point sharply. Two matching half circles sitting along one line. Notice that no length has been stated anywhere. None.

4:23So you choose one. Say the whole line is eight centimetres. That is allowed. And twelve would have been just as allowed, and so would three. But watch what happens the moment you commit. Eight forces X to the halfway mark at four, and forces the compass open to two. Choose twelve instead and X sits at six and the opening is three. Different numbers, exactly the same shape. The choice is free. The consequences are not. That is the difference between a length the problem gives you and a length the problem leaves to you.

5:03Now a kind of mark that surprises people the first time. This is an eye: two arcs meeting at both ends, one bulging up and one bulging down. To draw it you need two compass positions, one above the shape and one below. So draw two faint horizontal lines, one above and one below, and put the tips on them. Swing the upper arc. Swing the lower arc. And now rub the two faint lines out, because they were never part of the eye.

5:35They were there to locate something, and once it was located their job was finished. Extra lines like that are not mistakes. They are part of the method, and a construction is allowed to contain marks that get discarded. All of that planning happens somewhere, and it is worth being deliberate about where. It happens on a rough diagram. A rough diagram is a sketch that is deliberately not to scale, and carries no measurements at all.

6:08Here is one for a rectangle made of two identical squares. The proportions are wrong on purpose. What it does carry is the structure: which lengths have to be equal, marked with ticks, and which angles are known. That is enough to work out the order, and working it out costs nothing here, because nothing on this sketch has to be accurate. A rough diagram is not a bad drawing. It is a different tool doing a different job.

6:41Now the payoff, on a figure where the ordering question is the whole task. A house: two walls, a pointed apex above them, and a curved roof. Try to start with the apex and you cannot, because you do not yet know where it goes. So: the walls, which need nothing. Then two arcs swung from the tops of the walls. The apex is where those two arcs cross — so it needs the arcs.

7:08And the roof joins the apex back to the walls, so it needs both. Count the orders that work. Twenty-four arrangements of four things, and exactly one of them is drawable. Not one that is neater. One that is possible. And it is worth seeing what a figure with no valid order would look like. Suppose part P can only be drawn once you have Q, and Q can only be drawn once you have P.

7:39Then nothing needs nothing. There is nowhere to start, and no order works at all. You cannot draw that figure, and no amount of care with the instruments will help. Which turns the whole thing into one habit worth having. Before drawing, ask of every part: what does this need? Anything with an empty answer is where you begin. One last thing, and it is the step people skip. A construction is not finished when you draw the last line. It is finished when you check it.

8:14Suppose you had to build a rectangle. You measure, and the opposite sides come out equal. Done? Not yet. Here is a shape with both pairs of opposite sides equal that is leaning right over. Equal sides passed. Square corners failed. It is not a rectangle. So checking one condition is not checking the construction. You check every condition the figure was supposed to satisfy. Break it into parts, ask what each part needs, start where nothing is needed, and check at the end.

8:51Next time: what makes a rectangle a rectangle, and how to build one from the ground up.