The Logarithmic Spiral

Growth without changing shape

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• Logarithmic spiral $r = ae^{b\theta}$: the same growth factor every turn.
• Radii cross it at a constant angle $\alpha$ with $\cot\alpha = b$.
• Scaling it equals rotating it: perfect self-similarity.

Steps

1. The marvellous spiral
2. Growth per turn
3. A golden spiral
4. Zoom or rotate?
5. A nautilus
6. Moths and flames