Constructing 60° from an equilateral triangle, and the arches built on it

0:00Look at an arch, any arch, and ask how you would draw it exactly. Not roughly. Exactly, and twice, so that the second one matches the first. Here are two of them, and neither is the shape people expect. This one carries three lobes across the top, like a clover leaf turned on its side. This one comes to a point, with a corner where the two curves meet. Neither of them is a half circle sitting on the opening.

0:31And a curve is the one thing a hand cannot repeat. So before any of it is cut in stone, it is drawn flat. And the drawing does not begin with the curve. It begins with a few straight lines that the finished arch will hide completely. Those lines are the skeleton, and everything exact about the arch lives in them. Pin the skeleton with equal lengths and equal angles, and anyone can rebuild it anywhere.

1:02Then hang the curves on it. Which means one drawing holds two very different kinds of decision, and they are worth keeping apart. Take the clover one first. Here is its skeleton, with the curves stripped off. A straight base, with a corner at each end. From each corner, an arm leans inward and upward. Two conditions hold it together, and only two. The angle at the left end equals the angle at the right end.

1:34And the two arms are the same length as each other. The angles go down first, copied from one end to the other rather than measured, and only then are the arms stepped off. Now look at what those two conditions did on their own. The tops of the two arms came out level with each other. The whole figure folds onto itself down the middle of the base. Nobody measured for symmetry. Equal angles and equal arms delivered it.

2:05Even the two long crossing lines came out equal, and nobody asked for those at all. That level top is the gap the big middle lobe has to cross. Here the base is seventy-eight wide and the arms are twenty-five, which leaves a gap of forty-eight. Both conditions are doing work. Drop either one and the figure comes apart. Keep the two angles equal, but make one arm longer than the other.

2:36Now one arm top is higher than the other, and the fold down the middle fails. Or keep the arms equal, and tilt one of the angles. The same thing happens. The tops sit at different heights and there is no mirror. Equal angles on their own are not enough, and equal arms on their own are not enough. Together they are exactly enough, which is why those are the two things written on the skeleton.

3:05Now the curves, and here something changes. The big lobe has to start at one arm top and finish at the other. That fixes two points on it, and two points do not fix an arc. Set the centre just below the gap and the radius comes out at twenty-five, and the lobe rises to thirty-eight. Push the centre lower and the radius grows to twenty-six, then thirty, then forty, while the lobe gets flatter every time, topping out at thirty-six, then thirty-two, then twenty-eight.

3:41Each of those passes through both arm tops. Each of them is a real arch. The one thing not allowed is a radius under twenty-four, which cannot reach both ends at once. So the skeleton is exact and the curve is a choice, and there is no single correct radius to go hunting for. The pointed one is built differently, and more simply. Here is the opening it has to cross, and nothing else yet.

4:12Put the compass point on one end of that opening, and open it all the way to the other end. Swing a long arc up and over. Now the same again from the other end, with the opening untouched. The two arcs cross once above the middle, and that crossing is the top of the arch. Each arc already runs from a foot up to that crossing, so there is nothing left to draw.

4:39Two things about that shape are worth stopping on. A half circle on this same opening would rise to half of it, and stop. The pointed one goes well past that, standing on exactly the same two feet. And the top is a genuine corner rather than a smooth dome, because the two curves arrive there from different directions. Now look at what you actually drew. The opening across the bottom, and a reach from each end of it up to the top.

5:11Three lengths, all the same, because one compass opening made every one of them. Every construction up to here has grown out of a single angle: the right angle. It falls out of the line that cuts a segment in half and stands square to it. Halve that and you have forty-five. But there is a whole family of shapes those cannot reach. A triangle with three equal sides. A six-sided ring. A flower with six petals.

5:43All of them want sixty degrees, and nothing built so far produces one. And sixty turns out to be the cheapest angle you will ever construct. Here is the ray the angle will sit on, and here is the corner it starts from. Put the compass point on the corner and open it to anything you like. Swing an arc. It cuts the ray, and that cut becomes your second centre.

6:11Do not change the opening. Move the point onto that cut and swing again. The two arcs meet at a point above the ray. Rule a line from the corner out through the place where they met. That angle is sixty degrees, and nothing was measured anywhere in it. Why sixty, though, and why exactly sixty? Look at the triangle you built without meaning to. Corner to cut: that is the opening.

6:40Corner to the meeting point: the opening again, because the first arc put it there. Cut to the meeting point: the opening once more, because the second arc put it there. Three sides the same length. And a triangle with three equal sides has three equal angles. Three equal angles sharing a half turn between them, and a half turn split three ways is sixty. One thing has to be right, and it is not the size of the opening.

7:13The opening can be anything. Tiny, enormous, it makes no difference to the answer. What matters is that the second swing uses the same one as the first. Change it, and the third side stops matching the other two, and the angle moves with it. Open the second arc to twice the first, and the triangle flattens onto the ray completely. Open it wider than that, and the two arcs never meet, so there is nothing to rule a line to.

7:46One opening, used twice. That is the entire cost of sixty degrees. And once you have sixty, more of it comes free. Sit it on a straight line, and the other side of that line is a hundred and twenty. Halve the sixty and you have thirty. Halve that and you have fifteen. Which is a good moment to look back at how few things all of this started from.

8:12A right angle, and a sixty. Halve them, add them, take one from the other, and you reach forty-five, and a hundred and twenty, and thirty, and fifteen, and a great many more besides. You never land on forty. Or on fifty, or on seventy. You can creep towards forty for ever, and the miss keeps halving, from two and a half degrees to one and a quarter to a little over half a degree, and it never closes.

8:42One opening, swung twice. That is where all of it started.