Spirals: Archimedean, logarithmic and Fermat

r = aθ moves outward at a constant rate as it turns, so the gap between neighbouring turns is always 2πa. It is the shape of a rolled carpet, the groove on a vinyl record and a coiled rope. Archimedes used it around 225 BC to trisect angles and square the circle, problems impossible with ruler and compass alone.

r = ae^(bθ) grows by the same factor, e^(2πb), on every turn, so it looks the same at every scale. It also crosses every ray from the centre at the same angle. Jakob Bernoulli was so taken with it that he asked for it on his tombstone with the words “Eadem mutata resurgo” (changed, I rise again the same). Nautilus shells, the arms of spiral galaxies and the flight path of a moth circling a light all approximate it.

Fermat’s spiral r² = a²θ has two arms, r = ±a√θ. The area between successive turns is constant. Placing seeds along it at successive multiples of the golden angle, about 137.5°, reproduces the packing of seeds in a sunflower head, a model proposed by H. Vogel in 1979.