The Logarithmic Spiral

The logarithmic spiral is the curve that grows without changing shape. Nautilus, sunflower, galaxy arm, hurricane eye — they share a single equation because they share a single constraint: grow at a rate proportional to current size. This chapter sketches the proof that r = a·e^(bθ) is the unique planar curve invariant under similarity transformations centered at the origin.

Forward proof sketch: require self-similarity under scaling; derive the ODE dr/dθ = b·r; integrate; observe that the solution is unique up to the constants a and b.

Keynote phrase: the shell does not grow larger — it grows itself larger.

Witnesses queued: Jakob Bernoulli (who asked for the spiral on his tombstone), D’Arcy Thompson, the unnamed nautilus.

Ouroboros hook: closes on Bernoulli’s tombstone, whose engraver — unable to cut a true logarithmic spiral — cut an Archimedean one instead. The error is the proof of the rule.