Bisecting an angle, and halving 90° to get 45°

0:00Here is a figure worth wanting: eight narrow leaves, all the same, radiating from one centre. It is the kind of thing you would find carved into a door, or laid into a floor. Now try to draw it, and the trouble starts immediately. Not with the leaves. The leaves are arcs, and you can draw arcs. The trouble is where to put them. Eight things spaced evenly around a point is a demand about angles, and you have no protractor.

0:33So strip the picture back to what has to be built first. Under a design like this there are always some lines that never appear in the finished thing. You draw them to place everything, and then you rub them out. Here they are four straight lines, all crossing at the centre. Four lines through one point give you eight rays coming out of it, because each line runs both ways.

1:00So eight arms do not mean eight separate constructions. They mean four lines. Which is already a better problem than the one we started with. How far apart do those rays have to be? Go all the way round the centre and you have turned through three hundred and sixty degrees. The design cuts that turn into eight equal pieces. So divide. Three hundred and sixty, into eight. Forty-five degrees between one arm and the next, every time, all the way round.

1:34That number was not chosen. It was forced, by the decision to have eight arms. And now the whole design has come down to one question: how do you build an angle of forty-five degrees? You cannot. Not directly. A compass draws circles. Nothing about a circle hands you forty-five degrees. But look at the number for a moment. Forty-five is half of ninety. And ninety you can already build, anywhere you like, out of nothing but a compass and a straight edge.

2:11So the design does not need a way to make forty-five. It needs a way to cut something in half. Which is a much older kind of question, and one you have answered before. State it in general, because doing it only for ninety would be a waste. You are handed an angle. Any angle, any size, drawn by somebody else. Two arms, meeting at a corner. Call the corner O.

2:39You want the ray from O that splits the angle into two equal halves. No measuring. No reading anything off a scale. Just a compass and a straight edge, and an answer that is exact rather than close. Think about what would settle it, before thinking about how to draw anything. Suppose the answer ray is already there, and pick any point on it. Call that point C. Now there are two triangles sitting at the corner, sharing the side from O to C.

3:13One on each side of it. If those two triangles were congruent, matching part for part, then the two angles at O would have to match too. And two matching angles that add up to the whole thing are its two halves. So the drawing problem has turned into a different one. Build two triangles that must be congruent. Three matching sides are enough to force it. That is the strongest of the congruence conditions.

3:44One of the three is free: the side from O to C is in both triangles. It is the same side. So you need two more, and you have to manufacture them. Start at the corner. Open the compass, and cut both arms at the same distance from O. Call those two marks A and B. Now O to A and O to B are equal, because they are the same compass opening.

4:11One pair down. And something has just happened that is easy to walk straight past. Look at what is on the paper now. Two marks, and a point that is equally far from both of them. You have seen that before. The set of points equally far from two marks is a line: the one that cuts the segment between them in half and crosses it square. So the corner O is already on that line. Not near it. On it.

4:42And here is the thing worth stopping for. That line is the answer. The line that halves the angle is the same line that halves the segment from A to B. Which means you are not learning a new construction at all. You are finishing one you have already built. A line needs two points, and you are holding one of them. So the arcs owe you exactly one more.

5:08Open the compass again, to any width you like, as long as it is the same from both marks and long enough to reach. It does not have to be the width you used for A and B. In fact that width is the one to avoid, because one of the two crossings it gives you is the corner itself, which you already had. Strike an arc from A and an arc from B. Where they cross is C.

5:36C is equally far from A and from B, because both arcs had one radius. There is the second pair of sides. Three matching sides, so the triangles are congruent, so the two angles at O are equal. And the halving is not something you did. It fell out. So the whole thing is three steps, and it works on any angle you are handed. One: cut both arms at the same distance from the corner. Two: equal arcs from those two marks, crossing at C.

6:11Three: draw the line from the corner through C. Done. Now run it on a right angle. Two marks, two arcs, one line, and the ninety has become two angles of forty-five degrees each. Exact, not measured. Nobody read a scale at any point. Do it once more on one of those halves and you have twenty-two and a half degrees, which is as deep as the eight-armed design ever needs to go.

6:42Here is a question that unsettles people, and it should. What if you strike the two arcs the other way, so that they cross out past the corner instead of between the arms? Try it. Same two marks, bigger opening, arcs crossing on the far side. You get a point out there beyond O, and if you draw the line from the corner through it, it is the same line as before.

7:08Which should not be a surprise any more, because both crossings are equally far from A and B, and so is the corner. All three are on that one line, so there is only one line to draw. But say the next part carefully, because it is where the easy answer goes wrong. The line is the same. The ray pointing out to the far side is not inside your angle at all. It is halving the angle opposite.

7:38One more question, and this one has a limit hiding in it. Halving is now free. So what can you reach? From a right angle: forty-five, then twenty-two and a half, then eleven and a quarter, and on down. Add a second starting angle, the sixty degrees a triangle with three equal sides hands you, and you can add and subtract as well as halve. That opens up a lot. Fifteen degrees, seven and a half, sixty-seven and a half.

8:10Now try for sixty-five point five degrees, and you will not get there. Not ever, by any route. You can get close. Sixty degrees plus a right angle halved four times is sixty-five point six two five, which misses by an eighth of a degree. Halve again and you halve the miss, and again, and again, and it never once lands. Which leaves you with two things to build, and the tools to build both.

8:42The first is the figure we started with. Two lines square to each other, halve both of the right angles, and the four lines are there. Then eight arcs, two to a leaf, and rub out the skeleton. The second is a square with four leaves inside it, meeting at the centre and pointing at the corners. Each one is bounded by two arcs, and the arcs are centred on the middles of the sides.

9:09Ask for them as large as the square allows and the radius stops being yours to choose: any shorter and they do not reach the centre, any longer and they break out of the square. That is what the halving bought. Not a fact to remember. A design you can actually draw.