Logarithmic Spiral
SpiralsThe self-similar spiral r = a·e^{bθ} that crosses every radius at the same angle — nature’s growth curve.
Adjust parameters
Equation
Open the Creator with a ready-made animation brief for this curve — review the scene plan and Manim code before rendering.
What is the Logarithmic Spiral?
The logarithmic (or equiangular) spiral r = a·e^{bθ} multiplies its radius by a fixed factor with every turn, instead of adding a fixed amount. The result is perfect self-similarity: zooming in or out gives back the same spiral, only rotated. Jacob Bernoulli was so taken with the property that he asked for the curve on his tombstone with the motto “eadem mutata resurgo” — though the stonemason mistakenly carved an Archimedean spiral.
Its defining feature is the constant angle between the curve and every ray from the center, with cot of that angle equal to b. This is why hawks approach prey along one and why the nautilus shell, which grows without changing shape, follows it.
Key properties
- Equiangular: the tangent makes a constant angle α with every radius, cot α = b.
- Self-similar: scaling by e^{2πb} equals rotating by one turn.
- Each turn multiplies the radius by e^{2πb}.
- Total arc length from the pole to angle θ is finite: r(θ)·√(1+b²)/b.
- Special case b → 0 degenerates to a circle.
Where you'll see it
Nautilus shells, hurricane cloud bands, spiral galaxies, hawk flight paths, and low-noise spiral antennas.